Numerical
Formulas and
Tables
1234
12345
2
Table of Contents
Mathematical formulas………………………………………………………………….International System of Units (SI)……………………….................…….................Formulas of Dynamics……………………………………………..................…….....Table for Weight of Steel Bar………………………………………....……................List of Elements by Symbol and Specific Gravity………………………...............
Hardness Comparison Table......................................................................................Comparative Table of Gear Pitch...............................................................................Span Measurement Over k Teeth of Standard Spur Gear = 20° ...........................Span Measurement Over k Teeth of Standard Spur Gear = 14.5° ..........................Charts Indicating Span Measurement Over k Teeth of Spur and Helical Gears ..................Involute Function Table…………………………..............................…….................
34689
TABLE OF NUMBERS
NUMERICAL EXPRESSION, UNIT AND OTHER DATA
101112141617
56
3
Numerical Formulas and Tables
NUMERICAL EXPRESSION, UNIT AND OTHER DATA
Function y = f ( x ) and derivative y' = f ' ( x ) Trigonometric functions sin, cos, tan
cos θ = , sin θ = , tan θ =
sin2 θ + cos2 θ = 1, tan θ =
sin ( α ± β ) = sin α cos β ± cos α sin βcos ( α ± β ) = cos α cos β ± sin α sin β
The relationship between rectangular coordinates
(x, y)and polar coordinates (r, θ)
x = rcosθ , y = rsinθ ,
tan θ =
r 2 = x 2 + y 2
The equation for the straight line which passes the point (a, b), with an inclination m
( y − b ) = m ( x − a )
The equation for the circle, the center of which is the point (a, b) , with the radius r
( x − a ) 2 + ( y − b ) 2 = r 2
The root of the quadratic equation
y = ax 2 + bx + c = 0 (a 0)
x =
Function y = f ( x ) and derivative y' = f ' ( x )
1 Mathematical formulas
y = f ( x )
cos x
sin x
tan x
tan x − x NOTE 1.
y' = f ' ( x )
− sin x
cos x
tan2 x
y = f ( x )
sin2 x
sin3 x
y' = f ' ( x )
2 sin x cos x
3 sin2 x cos xNOTE 1. Involute function
An example of the application of the derivativeNewton's law
Introduced here is the method to obtain an approximate value of x 0 when the value of f (x0) is given in the involute function
f ( x ) = tan x − x
First, with an initial value x1
optionally chosen, obtain an approximate value x2 from:
x2 = x1 −
where f ' ( x1 ) = tan2 x1
Then, when the difference between the approximate value f ( x2 ) and the given f ( x0 ) is large, obtain an approximate value x3 using the same method.
x3 = x2 −
The more accurate value of x0 can be obtained by the repeated calculation using this method. The involute function, inv α, is frequently used in the calculation of gearing. Therefore, the Newton's law is really useful.
rx
ry
xy
cos θsin θ
xy
2a− b ± √ b2 − 4ac
tan2 x1
f ( x1 ) − f ( x0 )
tan2 x2
f ( x2 ) − f ( x0 )
cos2 x1
cos x1
sin x1
tan x1
cos xtan x
sin x tan x1
−
sin2 x1
−
x
yr
θ
x
y
rθ
P( x , y )
x
y
P( a , b )
x
y
P( a , b )
r
x
y
x1
y = f (x)
y
x
x
y
x0
x2 x10
f (x1)
f (x2)f (x0)
x2
f (x)
4
Numerical Formulas and Tables
square metrecubic metremetre per secondmetre per second squaredreciprocal metrekilogram per cubic metreampere per square metreampere per metremole per cubic metrecubic metre per kilogramcandela per square metre
SI supplementary unitQuantum
Physical quantum
2 International System of Units (SI)
SI Base unit
Definition of SI base unit
Length
Mass
Time
Current
Temperature
Amount of substance
Luminous intensity
Base unit
Standard International unit
meter
kilogram
second
ampere
kelvin
mole
candela
Definition
The length equal to the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second.
The mass equal to the mass of the international prototype kilogram stored at Sèvres, France.
The fundamental unit of time equal to 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of an atom of cesium-133.
One coulomb of electrical charge (6.24 x 1018 charge carriers) moving past a specific point in one second.
A unit of absolute temperature equal to 1/273.16 of the absolute temperature of the triple point of water.
An mount of substance of a system which contains as many elementary units as there are atoms of carbon in 0.012 kilogram of the pure nuclide carbon-12.
The magnitude of an electromagnetic field, in a specified direction, that has a power level of 1/683 watt (1.46 x 10-3 W) per steradian at a frequency of 540 terahertz (540 THz or 5.40 x 1014 Hz).
SI supplementary unit
Angle
Solid angle
Name
radian
steradian
Symbol
rad
sr
Definition
The angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle. The solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area r2.
SI derived unit
Quantum
AreaVolumeSpeedAccelerationWavenumberMass densityElectric current densityMagnetic field strengthAmount concentrationSpecific volumeLuminance
SI derived unit
Name Symbol
m2
m3
m/sm/s2
m-1
kg/m3
A/m2
A/mmol/m3
m3/kgcd/m2
5
Numerical Formulas and Tables
1kW = 1.01972×10 2 kgf m/s1kgf m/s = 9.80665×10 − 3 kW
Prefix
Prefix (Symbol)
yotta (Y)
zetta (Z)
exa (E)
peta (P)
tera (T)
giga (G)
mega (M)
kilo (k)
hecto (h)
deca (da)
Factor
1024
1021
1018
1015
1012
1092
1062
1032
1022
1012
Prefix (Symbol)
desi (d)
centi (c)
milli (m)
micro (n)
nano (n)
pico (p)
femto (f)
atto (a)
zepto (z)
yocto (y)
Factor
10 − 10
10 − 20
10 − 30
10 − 60
10 − 90
10 − 12
10 − 15
10 − 18
10 − 21
10 − 24
Conversion factors for SI (International System of Units)
Name
Force
Stress
WorkEnergy
WorkPower
Equivalents
1N = 1.01972×10 − 1 kgf
1kgf = 9.80665N
1MPa or N/mm2 = 1.01972×10 − 1 kgf/mm2
1kgf/mm2 = 9.80665MPa or N/mm2
1J = 2.77778×10 − 7 kW h = 1.01972×10 − 1 kgf m
1kgf m = 9.80665J, 1kW h = 3.600×106 J
Newton: The unit of force in the meter-kilogram-second system, equal to the force which will impart an acceleration of 1 meter per second squared to the International Prototype kilogram mass.Kilogram force: A unit of force equal to the weight of a 1-kilogram mass at a point on the earth's surface where the acceleration of gravity is 9.80665 m/s22.
6
Numerical Formulas and Tables
ω = = 2π n
Denomination
Denomination
Force
Moment of force (Torque)
Power (Dynamics)
Symbol and/or
formula
F
T = F r
P = FvLinear motion
P = TωRotational
motion
SI Conversion factors
1 kgf = 9.80665 N
1 kgf m = 9.80665 N m
When power is P ( kW ), force is F ( kgf ), velocity is v ( m/s ),
P = Fv ( kgf m /s) = 9.80665 Fv (W) = (kW)
When power is P ( kW ), torque is T ( kgf m ), rotational speed is n ( rpm ), P = Tω ( kgf m/s ) = 9.80665 T (W) = (kW)
Gravitational system of units
N (newton)
kgf
N m
kgf m
W
kgf m/s
W
kgf m/s
3 Formulas of Dynamics
Force, torque and dynamics
NOTE 1. Unit of dynamics 1W = 1 × 10−3 kW = 1J/S = 1N m/s
Comparison between linear and rotational motion
Displacement and angular displacementVelocity and angular velocityAcceleration and angular acceleration
Equation of motion
Work
Kinetic energy
Power(dynamics)
Linear motion
Symbol and/or formula
S
v =
α =
F = mα
E = Fs
E = mv2
P = Fv
Unit
m
m/s
m/s2
N
J
J
W
Rotational motion
Symbol and/or formula
θ
ω =
T = Iω
E = Tθ
E = Iω2
P = Tω
Unit
rad
rad/s
or 1/s
rad/s2
or 1/s2
kg m2/s2
or N m
J
J
W
NOTES 1. Radian (rad), the unit of plane angle θ, is dimesionless.
2. The unit of time( t) is second (s). 3. The unit of rotational speed (n) is 1/s (rps:
revolution per second) 4. The unit of mass(m) is kg.
5. 1N (newton) = 1kg m/s2
1N (newton) is the force which will impart an acceleration of 1 meter per second (1m/s2) squared to the International Prototype Kilogram mass (1kg).
6. The unit of "moment of inertia I = mk2" is kg m2. 7. The unit of work and kinetic energy is the same. 1J = 1N m
102Fv
30πn
974Tn
.
.
dtdS
dtdv
21
dtdω
dtdθ
21
NOTE 2 NOTE 3
NOTE 1
NOTE 6NOTE 5NOTE 4
NOTE 7
7
Numerical Formulas and Tables
Gravitational system of units kgf ms2
Denomination
Moment of inertia mk2 (kg m2) or k2 (kgf ms2)
Mass point
Pole
Plate
Cylinder
Hollow cylinder
Shape Rotation axis
y — y
y 1 — y 1
y 2 — y 2
y — y
z — z
y — y
z — z
y — y
Moment of inertia Unit
SI kg m2
mr 2
m
m
m
m
m
m
m
r 2
NOTE 1. m : Mass W : Weight g : Acceleration of gravity = 9.80665m/s2
NOTE 2. GD2 = 4gI (kgf m2)
12l 2
3l 2
12a2 + b2
12a2
2r 2
4r 2
2r 1
2 + r 22
gW
gW
12l 2
gW
3l 2
gW
12a2 + b2
12a2
gW
2r 2
gW
4r 2
gW
2r 1
2 + r 22
gW
gW
y
y
r
y
yr1
r2
y
y
r
z
z
y
y
b
az
z
y2 y1
y2 y12l
2l
8
Numerical Formulas and Tables
0.101.213.557.12
11.93
17.9625.2333.7343.4754.43
66.6380.0694.72
110.61127.73
146.09165.68186.50208.55231.84
256.36282.10309.08337.30366.74
397.42429.33462.47496.84532.45
569.28607.35646.65687.19728.95
771.95816.18861.64908.33956.25
1005.411055.801107.421160.271214.36
1269.671326.221384.001443.021503.26
1564.74
0.000.622.465.549.86
15.4022.1830.1839.4249.90
61.6074.5488.70
104.10120.74
138.60157.70178.02199.58222.38
246.40271.66298.14325.86354.82
385.00416.42449.06482.94518.06
554.40591.98630.78670.82712.10
754.60798.34843.30889.50936.94
985.601035.501086.621138.981192.58
1247.401303.461360.741419.261479.02
1540.00
Section
Round
Square
Hexagon
Dimension
(mm)
d
s
h
Weight of steel bar
for 1m (kgf/m)
0.00616d2
0.00785s2
0.00680h2
Material
Steel
Cast iron
Brass
Bronze
Specific gravity
(gf/cm3)
7.85
7.21
8.10
8.56
4 Table for Weight of Steel Bar
↑↓
↑↓
↑↓
Weight of round steel bar (kgf/m)
Dia.
0010203040
5060708090
100110120130140
150160170180190
200210220230240
250260270280290
300310320330340
350360370380390
400410420430440
450460470480490
500
0 1
0.010.752.725.92
10.36
16.0222.9231.0540.4251.01
62.8475.9090.19
105.71122.47
140.45159.67180.13201.81224.72
248.87274.25300.86328.70357.78
388.09419.63452.40486.40521.64
558.10595.80634.73674.90716.29
758.92802.78847.87894.19941.75
990.531040.551091.801144.291198.00
1252.951309.131366.541425.181485.06
1546.17
2
0.020.892.986.31
10.87
16.6623.6831.9341.4252.14
64.0977.2791.69
107.33124.21
142.32161.66182.24204.04227.08
251.35276.86303.59331.56360.75
391.19422.85455.74489.87525.23
561.82599.64638.69678.98720.50
763.25807.23852.45898.89946.57
995.481045.621097.001149.601203.44
1258.511314.821372.351431.121491.11
1552.34
3
0.061.043.266.71
11.39
17.3024.4532.8342.4453.28
65.3578.6693.19
108.96125.97
144.20163.67184.36206.29229.45
253.85279.47306.33334.42363.74
394.30426.08459.10493.35528.83
565.54603.49642.67683.08724.72
767.59811.70857.04903.60951.41
1000.441050.711102.201154.931208.89
1264.091320.511378.171437.061497.18
1558.54
4 5
0.151.393.857.55
12.47
18.6326.0334.6544.5155.59
67.9181.4796.25
112.27129.51
147.99167.71188.65210.83234.23
258.87284.75311.85340.19369.75
400.55432.59465.85500.35536.07
573.03611.23650.65691.31733.19
776.31820.67866.25913.07961.11
1010.391060.911112.651165.631219.83
1275.271331.951389.851448.991509.35
1570.95
6
0.221.584.167.98
13.03
19.3226.8335.5845.5656.77
69.2182.8997.80
113.94131.31
149.91169.75190.81213.11236.64
261.41287.40314.63343.09372.78
403.70435.86469.24503.86539.72
576.80615.11654.66695.44737.45
780.69825.17870.88917.82965.99
1015.391066.021117.891170.991225.32
1280.891337.681395.711454.971515.46
1577.18
7
0.301.784.498.43
13.61
20.0127.6536.5246.6357.96
70.5384.3299.35
115.62133.11
151.84171.80192.99215.41239.06
263.95290.07317.42346.00375.82
406.86439.14472.65507.39543.37
580.57619.01658.68699.59741.72
785.09829.68875.52922.58970.87
1020.401071.161123.151176.371230.82
1286.511343.431401.581460.961521.58
1583.42
8
0.392.004.838.90
14.19
20.7228.4837.4847.7059.16
71.8585.77
100.93117.31134.93
153.78173.86195.17217.72241.50
266.51292.75320.22348.93378.87
410.03442.44476.07510.94547.03
584.36622.92662.72703.74746.00
789.49834.21880.17927.35975.77
1025.421076.301128.411181.761236.34
1292.151349.191407.461466.971527.70
1589.67
9
0.502.225.189.37
14.79
21.4429.3338.4448.7960.37
73.1987.23
102.51119.02136.76
155.73175.94197.37220.04243.94
269.08295.44323.04351.87381.93
413.22445.74479.50514.49550.71
588.16626.85666.76707.91750.29
793.91828.75884.83932.14980.68
1030.451081.461133.691187.161241.86
1297.791354.961413.361472.991533.85
1595.94
EXAMPLE: Weight of round steel bar, diameter(128mm) and length(1m), is 100.93kgf.NOTE: Unit of diameter is millimeter (mm)
9
Numerical Formulas and Tables
Name
ZincAluminiumAntimonySulfurYtterbiumYttriumIridiumIndiumUraniumChlorineCadmiumPotassiumCalciumGoldSilverChlorineSiliconGermaniumCobaltOxygen
NaPbNbNiPtVPdBaAsFPuBeB
MgMnMo
IRaLiP
Name
BromineZirconiumMercuryHydrogenTinStrontiumCaesiumCeriumSeleniumBismuthThalliumTungstenCarbonTantalumTitaniumNitrogenIronTelluriumCopperThorium
Name
SodiumLeadNiobiumNickelPlatinumVanadiumPalladiumBariumArsenicFluorinePlutoniumBerylliumBoronMagnesiumManganeseMolybdenumIodineRadiumLithiumPhosphorus
Symbol
ZnAlSbS
YbYIrInUClCdKCaAuAgCrSiGeCoO
Symbol
BrZrHgHSnSrCsCeSeBiTlWCTaTiNFeTeCuTh
SymbolSpecific gravity
(20 ) g/cm3
07.133 (25°)02.69906.6202.0706.9604.4722.507.3119.0703.214×10−3
08.6500.8601.5519.3210.4907.1902.33 (25°)05.323 (25°)08.8501.429×10−3
03.1206.48913.54600.0899×10−3
07.298402.6001.903 (0°)06.7704.7909.8011.8519.302.2516.604.50701.250×10−3
07.8706.2408.9611.66
00.971211.3608.5708.902 (25°)21.4506.112.0203.505.7201.696×10−3
19.00 19.7201.84802.3401.7407.4310.2204.9405.000.53401.83
5 List of Elements by Symbol and Specific Gravity
Specific gravity
(20 )g/cm3
Specific gravity
( 20 )g/cm3
10
Numerical Formulas and Tables
10
1 Hardness Comparison Table
Approximate conversion values against Rockwell C hardness of steel materials
HRC
Rockwell
C
hardness
6867666564
6362616059
5857565554
5352515049
4847464544
4342414039
3837363534
3332313029
2827262524
23222120
(18)
(16)(14)(12)(10)(8)
(6)(4)(2)(0)
HV
Vickers
hardness
940900865832800
772746720697674
653633613595577
560544528513498
484471458446434
423412402392382
372363354345336
327318310302294
286279272266260
254248243238230
222213204196188
180173166160
HBBrinell hardness
10mm Ball Load 3000kgf
Standard ball
—————
—————
—————
—500487475464
451442432421409
400390381371362
353344336327319
311301294286279
271264258253247
243237231226219
212203194187179
171165158152
Hult-gren-ball
—————
———613599
587575561546534
519508494481469
455443432421409
400390381371362
353344336327319
311301294286279
271264258253247
243237231226219
212203194187179
171165158152
tungsten-carbide
ball
———739722
705688670654634
615595577560543
525512496481469
455443432421409
400390381371362
353344336327319
311301294286279
271264258253247
243237231226219
212203194187179
171165158152
HRA
A ScaleLoad 60kgf
braleindenter
85.685.084.583.983.4
82.882.381.881.280.7
80.179.679.078.578.0
77.476.876.375.975.2
74.774.173.673.172.5
72.071.570.970.469.9
69.468.968.467.967.4
66.866.365.865.364.7
64.363.863.362.862.4
62.061.561.060.5—
—————
————
Rockwell hardness NOTE 2.
HRB
B ScaleLoad 100kgfDia. 1/16in
Ball
—————
—————
—————
—————
—————
—————
——
(109.0)(108.5)(108.0)
(107.5)(107.0)(106.0)(105.5)(104.5)
(104.0)(103.0)(102.5)(101.5)(101.0)
100.0099.0098.5097.8096.7
095.5093.9092.3090.7089.5
087.1085.5083.5081.7
HRD
D ScaleLoad 100kgf
braleindenter
76.976.175.474.573.8
73.072.271.570.769.9
69.268.567.766.966.1
65.464.663.863.162.1
61.460.860.059.258.5
57.756.956.255.454.6
53.853.152.351.550.8
50.049.248.447.747.0
46.145.244.643.843.1
42.141.640.940.1—
—————
————
HS
Sho
re
hard
ness
9795929188
8785838180
7876757472
7169686766
6463626058
5756555452
5150494847
4644434241
4140383837
3635353433
3231292827
26252424
Tensile Strengthkgf/mm2
(N/mm2)Aprox. value
NOTE 1.
—————
—————
———
212 {2079}205 {2010}
199 {1952}192 {1883}186 {1824}179 {1755}172 {1687}
167 {1638}161 {1579}156 {1530}151 {1481}146 {1432}
141 {1383}136 {1334}132 {1294}127 {1245}124 {1216}
120 {1177}118 {1157}114 {1118}110 {1079}108 {1059}
105 {1030}102 {1000}100 {0981}097 {0951}095 {0932}
093 {0912}090 {0883}088 {0863}086 {0843}084 {0824}
082 {0804}080 {0785}079 {0775}077 {0755}075 {0736}
072 {0706}069 {0677}066 {0647}063 {0618}061 {0598}
059 {0579}056 {0549}054 {0530}053 {0520}
HRC
Rockwell C
Hardness
NOTE 2.
6867666564
6362616059
5857565554
5352515049
4847464544
4342414039
3837363534
3332313029
2827262524
23222120
(18)
(16)(14)(12)(10)(8)
(6)(4)(2)(0)
Approx. hardness
of
principal materials
S45C soft nitriding
SCM415 case hardening
SCM440 induction hardening
S45C induction hardening
SCM415 core hardness
S45C thermal refining
SCM440 core hardness
S45C core hardness
SUS303
REMARK: The boldfaced figures are based on ASTM E 140 Table 3 (SAE-ASM-ASTM)NOTE 1: Approximate values were determined from "Conversion Tables" of JIS Z 8413 and Z 8438. The units and values in parentheses{ } are based on International System of Units (SI). These are shown for reference. Note that 1N/mm2 = 1MPa.NOTE 2: The parenthesized values in the table are not used so frequently.
TABLE OF NUMBERS
11
Numerical Formulas and Tables
1.1.00531.01601.04721.05831.06391.09281.14001.14241.1545
1.19681.22761.251.25661.27001.32281.32991.39631.41111.4508
1.47841.51.57081.58751.59591.67551.69331.751.77321.7952
1.81431.93331.95381.99492.2.09442.09992.11672.18552.2166
2.252.28482.30912.34702.39362.49372.52.51332.54002.6456
2.65992.752.79252.82222.8499
02.956803.03.069103.141603.175003.191803.2503.324903.351003.5
03.590403.627103.628603.7503.866603.989804.04.188804.199804.2333
04.433204.504.569604.618204.693904.987305.05.026505.080005.3198
05.505.585105.644405.699806.06.138206.283206.3506.506.6497
07.07.180807.254207.257107.979708.08.377608.399708.466708.8663
09.09.236309.387809.974610.
010.0531010.16010.6395011.011.2889011.3995012.012.2764012.5664012.7000
013.013.2993014.014.5083014.5143015.015.9593016.016.7552016.9333
017.7326018.019.9492020.020.32022.022.7991023.024.025.
025.1327025.4026.026.5989028.029.030.031.4159031.9187032.
033.8663034.036.038.039.8683040.045.050.050.2955050.8
053.1977063.5000079.8019084.6667127.0000
Pitch
11
2 Comparative Table of Gear Pitch
Dimetral pitch
Diametral pitch
Diametral pitch
Pitch PitchModule
mModule
mModule
m
25.400025.265625.24.255024.23.873323.244422.281722.233922.
21.223120.690220.320020.212720.19.201919.098618.191418.17.5071
17.180616.933316.170116.15.915515.159515.14.514314.324014.1489
14.13.138213.12.732412.700012.127612.095812.11.622311.4592
11.288911.117011.10.822510.611710.185910.160010.106310.09.6010
09.549309.236409.095709.08.9127
8.59048.46678.27618.08518.7.95777.81537.63947.57987.2571
7.07447.00237.6.77336.56916.36626.35006.06386.04796.
5.72965.64445.55855.55.41135.09305.08005.05325.4.7747
4.61824.54794.54.45634.23334.13804.04254.3.90773.8197
3.62863.53723.50143.53.18313.17503.03193.02403.2.8648
2.82222.752.70562.54652.5400
2.52662.52.38732.30912.252.22822.11672.06902.02132.
1.95381.90991.81431.75071.751.69331.59151.58751.51601.5
1.43241.41111.27321.27001.251.15451.11411.10431.05831.0160
1.01061.0.97690.95490.90710.87590.84670.80350.79580.7938
0.750.74710.70560.66840.63660.63500.56440.50800.50530.5
0.47750.40.31830.30.2
in in in
3.14163.1⁄83.09223.2.96852.95282.7⁄82.75592.3⁄42.7211
2.5⁄82.55912.51332.1⁄22.47382.3⁄82.36222.1⁄42.22642.1654
2.1⁄82.09442.1.97901.96851.7⁄81.85531.79521.77171.3⁄4
1.73161.5⁄81.60791.57481.57081.1⁄21.49611.48431.7⁄16
1.4173
1.39631.3⁄81.36061.33861.5⁄16
1.25981.25661.1⁄41.23691.3⁄16
1.18111.14241.1⁄81.11321.1024
1.1⁄16
1.04721.02361.0.98950.98430.96660.94490.15⁄16
0.8976
0.7⁄80.86610.86580.83780.13⁄16
0.78740.78540.3⁄40.74800.7421
0.70870.6981011⁄16
0.68030.66950.62990.62830.5⁄80.61840.5906
0.57120.9⁄16
0.55660.55120.52360.51180.1⁄20.49480.48330.4724
0.44880.7⁄16
0.43310.43290.39370.39270.3⁄80.37400.37110.3543
0.34910.34010.33470.31500.3142
.5⁄16
.3092
.2953
.2856
.2783
.2756
.2618
.2559
.1⁄4
.2474
.2417
.2362
.2244
.2165
.2164
.2094
.1969
.1963
.3⁄16
.1855
.1772
.1745
.1575
.1571
.1546
.1428
.1378
.1366
.1309
.1257
.1⁄8
.1237
.1208
.1181
.1122
.1083
.1047
.1⁄10
.0984
.0982
.0928
.0924
.0873
.0827
.0787
.0785
.0698
.0628
.1⁄16
.0618
.0576
.0498
.0394
.0371
.0247
mm mm mm
79.79679.37578.54276.20075.40075.73.02570.69.85069.116
66.67565.63.83763.50062.83560.32560.57.15056.55155.
53.97553.19850.80050.26750.47.62547.12545.59845.44.450
43.98341.27540.84140.39.89838.10038.37.70136.51336.
35.46534.92534.55934.33.33832.31.91931.75031.41730.163
30.29.01728.57528.27528.
26.98826.59926.25.40025.13325.24.55224.23.81322.799
22.22522.21.99121.28020.63820.19.94919.05019.18.849
18.17.73317.46317.28017.16.15.95915.87515.70715.
14.50814.28814.13814.13.29913.12.70012.56812.27612.
11.39911.11311.10.99610.09.97509.52509.509.42609.3
08.86608.63908.508.07.980
7.9387.8547.57.2547.0697.6.6506.56.3506.283
6.1386.5.7005.55.4995.3205.4.9874.7634.712
4.54.4334.3.9903.9273.6273.53.4703.3253.193
3.1753.1423.0693.2.8502.7512.6602.5402.52.494
2.3572.3472.2172.1002.1.9951.7731.5961.5881.570
1.51.2561.0.9420.628
Numerical Formulas and Tables
4.4982
4.51224.52624.54024.55434.5683
4.58234.59634.61034.62434.6383
4.65234.66637.63247.64647.6604
7.67447.68857.70257.71657.7305
7.744510.710610.724610.738610.7526
10.766610.780610.794610.808610.8227
13.788813.802813.816813.830813.8448
13.858813.872813.886813.900816.8670
16.881016.895016.909016.923016.9370
16.951016.965016.979019.945219.9592
19.973219.987220.001220.015220.0292
12
z
12345
6789
10
1112131415
1617181920
2122232425
2627282930
3132333435
3637383940
4142434445
4647484950
5152535455
5657585960
z z z
121122123124125
126127128129130
131132133134135
136137138139140
141142143144145
146147148149150
151152153154155
156157158159160
161162163164165
166167168169170
171172173174175
176177178179180
181182183184185
186187188189190
191192193194195
196197198199200
201202203204205
206207208209210
211212213214215
216217218219220
221222223224225
226227228229230
231232233234235
236237238239240
k
22222
22222
22222
22333
33333
34444
44444
55555
55556
66666
66677
77777
k k
77888
88888
89999
99999
1010101010
1010101011
1111111111
1111111212
1212121212
1212131313
1313131313
1314141414
1414141414
1515151515
1515151516
1616161616
1616161717
1717171717
1717181818
1818181818
1819191919
1919191919
2020202020
2020202021
2121212121
2121212222
2222222222
2222232323
2323232323
2324242424
2424242424
2525252525
2525252526
2626262626
2626262727
2727272727
W W W W
20.043220.057223.023323.037323.0513
23.065323.079423.093423.107423.1214
23.135426.101526.115526.129526.1435
26.157526.171526.185626.199626.2136
29.179729.193729.207729.221729.2357
29.249729.263729.277729.291732.2579
32.271932.285932.299932.313932.3279
32.341932.355932.369935.336135.3501
35.364135.378135.392135.406135.4201
35.434135.448138.414238.428238.4423
38.456338.470338.484338.498338.5123
38.526341.492441.506441.520441.5344
41.548441.562441.576541.590541.6045
44.570644.584644.598644.612644.6266
44.640644.654644.668644.682647.6488
47.662847.676847.690847.704847.7188
47.732847.746847.760850.727050.7410
50.755050.769050.783050.797050.8110
50.825050.839053.805153.819253.8332
53.847253.861253.875253.889253.9032
53.917256.883356.897356.911356.9253
56.939456.953456.967456.981456.9954
59.961559.975559.989560.003560.0175
60.031560.045560.059660.073663.0397
63.053763.067763.081763.095763.1097
63.123763.137763.151766.117966.1319
66.145966.159966.173966.187966.2019
66.215966.229969.196169.210169.2241
69.238169.252169.266169.280169.2941
69.308172.274272.288272.302272.3163
72.330372.344372.358372.372372.3863
75.352475.366475.380475.394475.4084
75.422475.436475.450575.464578.4306
78.444678.458678.472678.486678.5006
78.514678.528678.542681.508881.5228
81.536881.550881.564881.578881.5928
3 Span Measurement Over k Teeth of Standard Spur Gear W ( x = 0 ) m = 1 ( α = 20° )
6162636465
6667686970
7172737475
7677787980
8182838485
8687888990
9192939495
96979899
100
101102103104105
106107108109110
111112113114115
116117118119120
k
Numerical Formulas and Tables
81.606881.620884.587084.601084.6150
84.629084.643084.657084.671084.6850
84.699087.665187.679187.693287.7072
87.721287.735287.749287.763287.7772
90.743390.757390.771390.785390.7993
90.813390.827490.841490.855493.8215
93.835593.849593.863593.877593.8915
93.905593.919593.933596.899796.9137
96.927796.941796.955796.969796.9837
96.997797.011799.977999.9919
100.0060
100.0200100.0340100.0480100.0620100.0760
100.0900103.0560103.0700103.0840103.0980
3434343434
3535353535
3535353536
3636363636
3636363737
3737373737
3737383838
3838383838
3839393939
3939393939
4040404040
4040404041
13
241242243244245
246247248249250
251252253254255
256257258259260
261262263264265
266267268269270
271272273274275
276277278279280
281282283284285
286287288289290
291292293294295
296297298299300
361362363364365
366367368369370
371372373374375
376377378379380
381382383384385
386387388389390
391392393394395
396397398399400
401402403404405
406407408409410
411412413414415
416417418419420
421422423424425
426427428429430
431432433434435
436437438439440
441442443444445
446447448449450
451452453454455
456457458459460
461462463464465
466467468469470
471472473474475
476477478479480
2727282828
2828282828
2829292929
2929292929
3030303030
3030303031
3131313131
3131313232
3232323232
3232333333
3333333333
3334343434
4141414141
4141414242
4242424242
4242434343
4343434343
4344444444
4444444444
4545454545
4545454546
4646464646
4646464747
4747474747
4747484848
4848484848
4849494949
4949494949
5050505050
5050505051
5151515151
5151515252
5252525252
5252535353
5353535353
5354545454
103.1120103.1260103.1400103.1540103.1680
106.1340106.1480106.1620106.1760106.1900
106.2040106.2180106.2320106.2460109.2120
109.2260109.2400109.2540109.2680109.2820
109.2960109.3100109.3240112.2910112.3050
112.3190112.3330112.3470112.3610112.3750
112.3890112.4030115.3690115.3830115.3970
115.4110115.4250115.4390115.4530115.4670
115.4810118.4470118.4610118.4750118.4890
118.5030118.5170118.5310118.5450118.5590
121.5250121.5390121.5530121.5670121.5810
121.5950121.6090121.6230121.6370124.6030
124.6170124.6310124.6450124.6590124.6730
124.6870124.7010124.7150127.6810127.6960
127.7100127.7240127.7380127.7520127.7660
127.7800127.7940130.7600130.7740130.7880
130.8020130.8160130.8300130.8440130.8580
130.8720133.8380133.8520133.8660133.8800
133.8940133.9080133.9220133.9360133.9500
136.9160136.9300136.9440136.9580136.9720
136.9860137.0000137.0140137.0280139.9940
140.0080140.0220140.0360140.0500140.0640
140.0780140.0920140.1060143.0720143.0860
143.1000143.1140143.1280143.1420143.1560
143.1700143.1840146.1510146.1650146.1790
146.1930146.2070146.2210146.2350146.2490
146.2630149.2290149.2430149.2570149.2710
149.2850149.2990149.3130149.3270149.3410
152.3070152.3210152.3350152.3490152.3630
152.3770152.3910152.4050152.4190155.3850
155.3990155.4130155.4270155.4410155.4550
155.4690155.4830155.4970158.4630158.4770
158.4910158.5050158.5190158.5330158.5470
158.5610158.5750161.5420161.5560161.5700
161.5840161.5980161.6120161.6260161.6400
161.6540164.6200164.6340164.6480164.6620
m = 1 ( α = 20° )
301302303304305
306307308309310
311312313314315
316317318319320
321322323324325
326327328329330
331332333334335
336337338339340
341342343344345
346347348349350
351352353354355
356357358359360
z z z zk k kW W W Wk
Numerical Formulas and Tables
4.59994.60524.61064.6160
4.62134.62674.63214.63744.6428
4.64824.65364.65894.66434.6697
4.67504.68044.68584.69117.7380
7.74347.74887.75417.75957.7649
7.77027.77567.78107.78637.7917
7.79717.8024
10.849310.854710.8601
10.865410.870810.876210.881510.8869
10.892310.897610.903010.908413.9553
13.960613.966013.971413.976713.9821
13.987513.992913.998214.003614.0090
14.014314.019717.066617.072017.0773
17.082717.088117.093417.098817.1042
17.109517.114917.120317.125620.1725
20.177920.183320.188620.194020.1994
20.204720.210120.215520.220820.2262
20.231623.278523.283823.289223.2946
23.300023.305323.310723.316123.3214
23.326823.332223.337523.342926.3898
26.395226.400526.405926.411326.4166
26.422026.427426.432726.438126.4435
26.448829.495729.501129.506529.5118
29.517229.522629.527929.533329.5387
14
12345
6789
10
1112131415
1617181920
2122232425
2627282930
3132333435
3637383940
4142434445
4647484950
5152535455
5657585960
121122123124125
126127128129130
131132133134135
136137138139140
141142143144145
146147148149150
151152153154155
156157158159160
161162163164165
166167168169170
171172173174175
176177178179180
181182183184185
186187188189190
191192193194195
196197198199200
201202203204205
206207208209210
211212213214215
216217218219220
221222223224225
226227228229230
231232233234235
236237238239240
22222
22222
22222
22222
22223
33333
33333
33444
44444
44445
55555
55555
55666
66666
66667
77777
77777
78888
88888
88889
99999
99999
910101010
1010101010
1010101011
1111111111
1111111111
1112121212
1212121212
1212121313
1313131313
1313131313
1314141414
1414141414
1414141515
1515151515
1515151515
1516161616
1616161616
1616161717
1717171717
1717171717
1718181818
1818181818
1818181919
1919191919
1919191919
2020202020
29.544029.549429.554829.560232.6070
32.612432.617832.623232.628532.6339
32.639332.644632.650032.655432.6607
32.666135.713035.718435.723735.7291
35.734535.739835.745235.750635.7559
35.761335.766735.772038.818938.8243
38.829738.835038.840438.845838.8511
38.856538.861938.867238.872638.8780
38.883441.930214.935641.941041.9464
41.951741.957141.962541.967841.9732
41.978641.983941.989345.036245.0416
45.046945.052345.057745.063045.0684
45.073845.079145.084545.089945.0952
45.100648.147548.152948.158248.1636
48.169048.174348.179748.185148.1905
48.195848.201248.206651.253451.2588
51.264251.269651.274951.280351.2857
51.291051.296451.301851.307151.3125
51.317954.364854.370154.375554.3809
54.386254.391654.397054.402354.4077
54.413154.418454.423857.470757.4761
57.481457.486857.492257.497557.5029
57.508357.513757.519057.524457.5298
60.576660.582060.587460.592860.5981
4 Span Measurement Over k Teeth of Standard Spur Gear W ( x = 0 ) m = 1 ( α = 14.5° )
6162636465
6667686970
7172737475
7677787980
8182838485
8687888990
9192939495
96979899
100
101102103104105
106107108109110
111112113114115
116117118119120
z z z zk k kW W W Wk
Numerical Formulas and Tables
91.662991.668391.673791.679091.6844
91.689891.695191.700591.705991.7113
91.716691.722094.768994.774294.7796
94.785094.790494.795794.801194.8065
94.811894.817294.822694.827997.8748
97.880297.885697.890997.896397.9017
97.907097.912497.917897.923197.9285
97.933997.9392
100.9860100.9920100.9970
101.0020101.0080101.0130101.0180101.0240
101.0290101.0340101.0400101.0450104.0920
104.0970104.1030104.1080104.1140104.1190
104.1240104.1300104.1350104.1400104.1460
60.603560.608960.614260.619660.6250
60.630360.635760.641163.688063.6933
63.698763.704163.709463.714863.7202
63.725563.730963.736363.741663.7470
66.793966.799366.804666.810066.8154
66.820766.826166.831566.836966.8422
66.847666.853066.858369.905269.9106
69.916069.921369.926769.932169.9374
69.942869.948269.953569.958969.9643
73.011273.016573.021973.027373.0326
73.038073.043473.048773.054173.0595
73.064973.070276.117176.122576.1278
15
2525252525
2525252525
2626262626
2626262626
2626272727
2727272727
2727272727
2828282828
2828282828
2828292929
2929292929
2929292930
241242243244245
246247248249250
251252253254255
256257258259260
261262263264265
266267268269270
271272273274275
276277278279280
281282283284285
286287288289290
291292293294295
296297298299300
361362363364365
366367368369370
371372373374375
376377378379380
381382383384385
386387388389390
391392393394395
396397398399400
401402403404405
406407408409410
411412413414415
416417418419420
421422423424425
426427428429430
431432433434435
436437438439440
441442443444445
446447448449450
451452453454455
456457458459460
461462463464465
466467468469470
471472473474475
476477478479480
2020202020
2020202121
2121212121
2121212121
2222222222
2222222222
2222222323
2323232323
2323232323
2424242424
2424242424
2424252525
3030303030
3030303030
3030313131
3131313131
3131313132
3232323232
3232323232
3232333333
3333333333
3333333334
3434343434
3434343434
3434353535
3535353535
3535353536
3636363636
3636363636
3637373737
3737373737
3737373738
3838383838
3838383838
3839393939
3939393939
76.133276.138676.144076.149376.1547
76.160176.165476.170876.176276.1815
79.228479.233879.239279.244579.2499
79.255379.260679.266079.271479.2767
79.282179.287582.334482.339782.3451
82.350582.355882.361282.366682.3719
82.377382.382782.388082.393482.3988
85.445785.451085.456485.461885.4671
85.472585.477985.483385.488685.4940
85.499485.504788.551688.557088.5624
88.567788.573188.578588.583888.5892
88.594688.599988.605388.610791.6576
104.1510104.1560107.2030107.2090107.2140
107.2190107.2250107.2300107.2360107.2410
107.2460107.2520107.2570107.2620110.3090
110.3150110.3200110.3250110.3310110.3360
110.3420110.3470110.3520110.3580110.3630
110.3680113.4150113.4210113.4260113.4310
113.4370113.4420113.4470113.4530113.4580
113.4640113.4690113.4740113.4800116.5270
116.5320116.5370116.5430116.5480116.5530
116.5590116.5640116.5700116.5750116.5800
116.5860119.6330119.6380119.6430119.6490
119.6540119.6590119.6650119.6700119.6750
m = 1 ( α = 14.5° )
301302303304305
306307308309310
311312313314315
316317338319320
321322323324325
326327328329330
331332333334335
336337338339340
341342343344345
346347348349350
351352353354355
356357358359360
z z z zk k kW W W Wk
Numerical Formulas and Tables
16
5 Charts Indicating Spanned Number of Teeth of Spur and Helical Gears (Maag's data)
Numerical Formulas and Tables
0.000014180.000014540.000014910.000015280.00001565
0.000016030.000016420.000016820.000017220.00001762
0.000018040.000018460.000018880.000019310.00001975
0.000020200.000020650.000021110.000021580.00002205
0.000022530.000023010.000023510.000024010.00002452
0.000025030.000025550.000026080.000026620.00002716
0.000027710.000028270.000028840.000029410.00002999
0.000030580.000031170.000031780.000032390.00003301
0.000033640.000034270.000034910.000035560.00003622
0.000036890.000037570.000038250.000038940.00003964
0.000040350.000041070.000041790.000042520.00004327
0.000044020.000044780.000045540.000046320.00004711
0.00004790
0001020304
0506070809
1011121314
1516171819
2021222324
2526272829
3031323334
3536373839
4041424344
4546474849
5051525354
5556575859
60
17
6 Involute Function Table inv α = tan α − α
2°
0.000047900.000048710.000049520.000050340.00005117
0.000052010.000052860.000053720.000054580.00005546
0.000056340.000057240.000058140.000059060.00005998
0.000060910.000061860.000062810.000063770.00006474
0.000065730.000066720.000067720.000068730.00006975
0.000070780.000071830.000072880.000073940.00007501
0.000076100.000077190.000078290.000079410.00008053
0.000081670.000082810.000083970.000085140.00008632
0.000087510.000088710.000089920.000091140.00009237
0.000093620.000094870.000096140.000097420.00009870
0.000100000.000101320.000102640.000103970.00010532
0.000106680.000108050.000109430.000110820.00011223
0.00011364
0.00011360.00011510.00011650.00011800.0001194
0.00012090.00012240.00012390.00012540.0001269
0.00012850.00013000.00013160.00013320.0001347
0.00013630.00013800.00013960.00014120.0001429
0.00014450.00014620.00014790.00014960.0001513
0.00015300.00015480.00015650.00015830.0001601
0.00016190.00016370.00016550.00016740.0001692
0.00017110.00017290.00017480.00017670.0001787
0.00018060.00018250.00018450.00018650.0001885
0.00019050.00019250.00019450.00019650.0001986
0.00020070.00020280.00020490.00020700.0002091
0.00021130.00021340.00021560.00021780.0002200
0.0002222
0.00022220.00022440.00022670.00022890.0002312
0.00023350.00023580.00023820.00024050.0002429
0.00024520.00024760.00025000.00025240.0002549
0.00025730.00025980.00026220.00026470.0002673
0.00026980.00027230.00027490.00027750.0002801
0.00028270.00028530.00028790.00029060.0002933
0.00029590.00029860.00030140.00030410.0003069
0.00030960.00031240.00031520.00031800.0003209
0.00032370.00032660.00032950.00033240.0003353
0.00033830.00034120.00034420.00034720.0003502
0.00035320.00035630.00035930.00036240.0003655
0.00036860.00037180.00037490.00037810.0003813
0.0003845
0.00038450.00038770.00039090.00039420.0003975
0.00040080.00040410.00040740.00041080.0004141
0.00041750.00042090.00042440.00042780.0004313
0.00043470.00043820.00044170.00044530.0004488
0.00045240.00045600.00045960.00046320.0004669
0.00047060.00047430.00047800.00048170.0004854
0.00048920.00049300.00049680.00050060.0005045
0.00050830.00051220.00051610.00052000.0005240
0.00052800.00053190.00053590.00054000.0005440
0.00054810.00055220.00055630.00056040.0005645
0.00056870.00057290.00057710.00058130.0005856
0.00058980.00059410.00059850.00060280.0006071
0.0006115
0.00061150.00061590.00062030.00062480.0006292
0.00063370.00063820.00064270.00064730.0006518
0.00065640.00066100.00066570.00067030.0006750
0.00067970.00068440.00068920.00069390.0006987
0.00070350.00070830.00071320.00071810.0007230
0.00072790.00073280.00073780.00074280.0007478
0.00075280.00075790.00076290.00076800.0007732
0.00077830.00078350.00078870.00079390.0007991
0.00080440.00080960.00081500.00082030.0008256
0.00083100.00083640.00084180.00084730.0008527
0.00085820.00086380.00086930.00087490.0008805
0.00088610.00089170.00089740.00090310.0009088
0.0009145
0.00091450.00092030.00092600.00093180.0009377
0.00094350.00094940.00095530.00096120.0009672
0.00097320.00097920.00098520.00099130.0009973
0.00100340.00100960.00101570.00102190.0010281
0.00103430.00104060.00104690.00105320.0010595
0.00106590.00107220.00107860.00108510.0010915
0.00109800.00110450.00111110.00111760.0011242
0.00113080.00113750.00114410.00115080.0011575
0.00116430.00117110.00117790.00118470.0011915
0.00119840.00120530.00121220.00121920.0012262
0.00123320.00124020.00124730.00125440.0012615
0.00126870.00127580.00128300.00129030.0012975
0.0013048
0.0013050.0013120.0013190.0013270.001334
0.0013420.0013490.0013570.0013640.001372
0.0013790.0013870.0013940.0014020.001410
0.0014170.0014250.0014330.0014410.001448
0.0014560.0014640.0014720.0014800.001488
0.0014960.0015040.0015120.0015200.001528
0.0015360.0015440.0015530.0015610.001569
0.0015770.0015860.0015940.0016020.001611
0.0016190.0016280.0016360.0016450.001653
0.0016620.0016700.0016790.0016880.001696
0.0017050.0017140.0017230.0017310.001740
0.0017490.0017580.0017670.0017760.001785
0.001794
0.0017940.0018030.0018120.0018210.001830
0.0018400.0018490.0018580.0018670.001877
0.0018860.0018950.0019050.0019140.001924
0.0019330.0019430.0019520.0019620.001972
0.0019810.0019910.0020010.0020100.002020
0.0020300.0020400.0020500.0020600.002069
0.0020790.0020890.0021000.0021100.002120
0.0021300.0021400.0021500.0021600.002171
0.0021810.0021910.0022020.0022120.002223
0.0022330.0022440.0022540.0022650.002275
0.0022860.0022970.0023070.0023180.002329
0.0023400.0023500.0023610.0023720.002383
0.002394
0.0023940.0024050.0024160.0024270.002438
0.0024490.0024610.0024720.0024830.002494
0.0025060.0025170.0025280.0025400.002551
0.0025630.0025740.0025860.0025980.002609
0.0026210.0026330.0026440.0026560.002668
0.0026800.0026920.0027030.0027150.002727
0.0027390.0027510.0027640.0027760.002788
0.0028000.0028120.0028250.0028370.002849
0.0028620.0028740.0028870.0028990.002912
0.0029240.0029370.0029490.0029620.002975
0.0029870.0030000.0030130.0030260.003039
0.0030520.0030650.0030780.0030910.003104
0.003117
3° 4° 5° 6° 7° 8° 9° 10° 11°
Numerical Formulas and Tables
0001020304
0506070809
1011121314
1516171819
2021222324
2526272829
3031323334
3536373839
4041424344
4546474849
5051525354
5556575859
60
0.0031170.0031300.0031430.0031570.003170
0.0031830.0031970.0032100.0032230.003237
0.0032500.0032640.0032770.0032910.003305
0.0033180.0033320.0033460.0033600.003374
0.0033870.0034010.0034150.0034290.003443
0.0034580.0034720.0034860.0035000.003514
0.0035290.0035430.0035570.0035720.003586
0.0036000.0036150.0036300.0036440.003659
0.0036730.0036880.0037030.0037180.003733
0.0037470.0037620.0037770.0037920.003807
0.0038220.0038380.0038530.0038680.003883
0.0038980.0039140.0039290.0039440.003960
0.003975
18
inv α = tan α − αInvolute Function Table
12°
0.0039750.0039910.0040060.0040220.004038
0.0040530.0040690.0040850.0041010.004117
0.0041330.0041480.0041640.0041800.004197
0.0042130.0042290.0042450.0042610.004277
0.0042940.0043100.0043270.0043430.004359
0.0043760.0043930.0044090.0044260.004443
0.0044590.0044760.0044930.0045100.004527
0.0045440.0045610.0045780.0045950.004612
0.0046290.0046460.0046640.0046810.004698
0.0047160.0047330.0047510.0047680.004786
0.0048030.0048210.0048390.0048560.004874
0.0048920.0049100.0049280.0049460.004964
0.004982
0.0049820.0050000.0050180.0050360.005055
0.0050730.0050910.0051100.0051280.005146
0.0051650.0051840.0052020.0052210.005239
0.0052580.0052770.0052960.0053150.005334
0.0053530.0053720.0053910.0054100.005429
0.0054480.0054670.0054870.0055060.005525
0.0055450.0055640.0055840.0056030.005623
0.0056430.0056620.0056820.0057020.005722
0.0057420.0057620.0057820.0058020.005822
0.0058420.0058620.0058820.0059030.005923
0.0059430.0059640.0059840.0060050.006025
0.0060460.0060670.0060870.0061080.006129
0.006150
0.0061500.0061710.0061920.0062130.006234
0.0062550.0062760.0062970.0063180.006340
0.0063610.0063820.0064040.0064250.006447
0.0064690.0064900.0065120.0065340.006555
0.0065770.0065990.0066210.0066430.006665
0.0066870.0067090.0067320.0067540.006776
0.0067990.0068210.0068430.0068660.006888
0.0069110.0069340.0069560.0069790.007002
0.0070250.0070480.0070710.0070940.007117
0.0071400.0071630.0071860.0072090.007233
0.0072560.0072800.0073030.0073270.007350
0.0073740.0073970.0074210.0074450.007469
0.007493
0.0074930.0075170.0075410.0075650.007589
0.0076130.0076370.0076610.0076860.007710
0.0077350.0077590.0077840.0078080.007833
0.0078570.0078820.0079070.0079320.007957
0.0079820.0080070.0080320.0080570.008082
0.0081070.0081330.0081580.0081830.008209
0.0082340.0082600.0082850.0083110.008337
0.0083620.0083880.0084140.0084400.008466
0.0084920.0085180.0085440.0085710.008597
0.0086230.0086500.0086760.0087020.008729
0.0087560.0087820.0088090.0088360.008863
0.0088890.0089160.0089430.0089700.008998
0.009025
0.0090250.0090520.0090790.0091070.009134
0.0091610.0091890.0092160.0092440.009272
0.0092990.0093270.0093550.0093830.009411
0.0094390.0094670.0094950.0095230.009552
0.0095800.0096080.0096370.0096650.009694
0.0097220.0097510.0097800.0098080.009837
0.0098660.0098950.0099240.0099530.009982
0.0100110.0100410.0100700.0100990.010129
0.0101580.0101880.0102170.0102470.010277
0.0103070.0103360.0103660.0103960.010426
0.0104560.0104860.0105170.0105470.010577
0.0106080.0106380.0106690.0106990.010730
0.010760
0.0107600.0107910.0108220.0108530.010884
0.0109150.0109460.0109770.0110080.011039
0.0110710.0111020.0111330.0111650.011196
0.0112280.0112600.0112910.0113230.011355
0.0113870.0114190.0114510.0114830.011515
0.0115470.0115800.0116120.0116440.011677
0.0117090.0117420.0117750.0118070.011840
0.0118730.0119060.0119390.0119720.012005
0.0120380.0120710.0121050.0121380.012172
0.0122050.0122390.0122720.0123060.012340
0.0123730.0124070.0124410.0124750.012509
0.0125430.0125780.0126120.0126460.012681
0.012715
0.0127150.0127500.0127840.0128190.012854
0.0128880.0129230.0129580.0129930.013028
0.0130630.0130980.0131340.0131690.013204
0.0132400.0132750.0133110.0133460.013382
0.0134180.0134540.0134900.0135260.013562
0.0135980.0136340.0136700.0137070.013743
0.0137790.0138160.0138520.0138890.013926
0.0139630.0139990.0140360.0140730.014110
0.0141480.0141850.0142220.0142590.014297
0.0143340.0143720.0144090.0144470.014485
0.0145230.0145600.0145980.0146360.014674
0.0147130.0147510.0147890.0148270.014866
0.014904
0.0149040.0149430.0149820.0150200.015059
0.0150980.0151370.0151760.0152150.015254
0.0152930.0153330.0153720.0154110.015451
0.0154900.0155300.0155700.0156090.015649
0.0156890.0157290.0157690.0158090.015849
0.0158900.0159300.0159710.0160110.016052
0.0160920.0161330.0161740.0162140.016255
0.0162960.0163370.0163790.0164200.016461
0.0165020.0165440.0165850.0166270.016669
0.0167100.0167520.0167940.0168360.016878
0.0169200.0169620.0170040.0170470.017089
0.0171320.0171740.0172170.0172590.017302
0.017345
0.0173450.0173880.0174310.0174740.017517
0.0175600.0176030.0176470.0176900.017734
0.0177770.0178210.0178650.0179080.017952
0.0179960.0180400.0180840.0181290.018173
0.0182170.0182620.0183060.0183510.018395
0.0184400.0184850.0185300.0185750.018620
0.0186650.0187100.0187550.0188000.018846
0.0188910.0189370.0189830.0190280.019074
0.0191200.0191660.0192120.0192580.019304
0.0193500.0193970.0194430.0194900.019536
0.0195830.0196300.0196760.0197230.019770
0.0198170.0198640.0199120.0199590.020006
0.020054
13° 14° 15° 16° 17° 18° 19° 20° 21°
Numerical Formulas and Tables
0.0200540.0201010.0201490.0201970.020244
0.0202920.0203400.0203880.0204360.020484
0.0205330.0205810.0206290.0206780.020726
0.0207750.0208240.0208730.0209210.020970
0.0210190.0210690.0211180.0211670.021217
0.0212660.0213150.0213650.0214150.021465
0.0215140.0215640.0216140.0216650.021715
0.0217650.0218150.0218660.0219160.021967
0.0220180.0220680.0221190.0221700.022221
0.0222720.0223240.0223750.0224260.022478
0.0225290.0225810.0226320.0226840.022736
0.0227880.0228400.0228920.0229440.022997
0.023049
19
inv α = tan α − αInvolute Function Table
0001020304
0506070809
1011121314
1516171819
2021222324
2526272829
3031323334
3536373839
4041424344
4546474849
5051525354
5556575859
60
22°
0.0230490.0231020.0231540.0232070.023259
0.0233120.0233650.0234180.0234710.023524
0.0235770.0236310.0236840.0237380.023791
0.0238450.0238990.0239520.0240060.024060
0.0241140.0241690.0242230.0242770.024332
0.0243860.0244410.0244950.0245500.024605
0.0246600.0247150.0247700.0248250.024881
0.0249360.0249920.0250470.0251030.025159
0.0252140.0252700.0253260.0253820.025439
0.0254950.0255510.0256080.0256640.025721
0.0257780.0258340.0258910.0259480.026005
0.0260620.0261200.0261770.0262350.026292
0.026350
0.0263500.0264070.0264650.0265230.026581
0.0266390.0266970.0267560.0268140.026872
0.0269310.0269890.0270480.0271070.027166
0.0272250.0272840.0273430.0274020.027462
0.0275210.0275810.0276400.0277000.027760
0.0278200.0278800.0279400.0280000.028060
0.0281210.0281810.0282420.0283020.028363
0.0284240.0284850.0285460.0286070.028668
0.0287290.0287910.0288520.0289140.028976
0.0290370.0290990.0291610.0292230.029285
0.0293480.0294100.0294720.0295350.029598
0.0296600.0297230.0297860.0298490.029912
0.029975
0.0299750.0300390.0301020.0301660.030229
0.0302930.0303570.0304200.0304840.030549
0.0306130.0306770.0307410.0308060.030870
0.0309350.0310000.0310650.0311300.031195
0.0312600.0313250.0313900.0314560.031521
0.0315870.0316530.0317180.0317840.031850
0.0319170.0319830.0320490.0321160.032182
0.0322490.0323150.0323820.0324490.032516
0.0325830.0326510.0327180.0327850.032853
0.0329200.0329880.0330560.0331240.033192
0.0332600.0333280.0333970.0334650.033534
0.0336020.0336710.0337400.0338090.033878
0.033947
0.0339470.0340160.0340860.0341550.034225
0.0342940.0343640.0344340.0345040.034574
0.0346440.0347140.0347850.0348550.034926
0.0349960.0350670.0351380.0352090.035280
0.0353520.0354230.0354940.0355660.035637
0.0357090.0357810.0358530.0359250.035997
0.0360690.0361420.0362140.0362870.036359
0.0364320.0365050.0365780.0366510.036724
0.0367980.0368710.0369450.0370180.037092
0.0371660.0372400.0373140.0373880.037462
0.0375370.0376110.0376860.0377610.037835
0.0379100.0379850.0380600.0381360.038211
0.038287
0.0382870.0383620.0384380.0385140.038590
0.0386660.0387420.0388180.0388940.038971
0.0390470.0391240.0392010.0392780.039355
0.0394320.0395090.0395860.0396640.039741
0.0398190.0398970.0399740.0400520.040131
0.0402090.0402870.0403660.0404440.040523
0.0406020.0406800.0407590.0408380.040918
0.0409970.0410760.0411560.0412360.041316
0.0413950.0414750.0415560.0416360.041716
0.0417970.0418770.0419580.0420390.042120
0.0422010.0422820.0423630.0424440.042526
0.0426070.0426890.0427710.0428530.042935
0.043017
0.0430170.0431000.0431820.0432640.043347
0.0434300.0435130.0435960.0436790.043762
0.0438450.0439290.0440120.0440960.044180
0.0442640.0443480.0444320.0445160.044601
0.0446850.0447700.0448550.0449390.045024
0.0451100.0451950.0452800.0453660.045451
0.0455370.0456230.0457090.0457950.045881
0.0459670.0460540.0461400.0462270.046313
0.0464000.0464870.0465750.0466620.046749
0.0468370.0469240.0470120.0471000.047188
0.0472760.0473640.0474520.0475410.047630
0.0477180.0478070.0478960.0479850.048074
0.048164
0.0481640.0482530.0483430.0484320.048522
0.0486120.0487020.0487920.0488830.048973
0.0490630.0491540.0492450.0493360.049427
0.0495180.0496090.0497010.0497920.049884
0.0499760.0500680.0501600.0502520.050344
0.0504370.0505290.0506220.0507150.050808
0.0509010.0509940.0510870.0511810.051274
0.0513680.0514620.0515560.0516500.051744
0.0518380.0519330.0520270.0521220.052217
0.0523120.0524070.0525020.0525970.052693
0.0527880.0528840.0529800.0530760.053172
0.0532680.0533650.0534610.0535580.053655
0.053751
0.0537510.0538490.0539460.0540430.054140
0.0542380.0543360.0544330.0545310.054629
0.0547280.0548260.0549240.0550230.055122
0.0552210.0553200.0554190.0555180.055617
0.0557170.0558170.0559160.0560160.056116
0.0562170.0563170.0564170.0565180.056619
0.0567200.0568210.0569220.0570230.057124
0.0572260.0573280.0574290.0575310.057633
0.0577360.0578380.0579400.0580430.058146
0.0582490.0583520.0584550.0585580.058662
0.0587650.0588690.0589730.0590770.059181
0.0592850.0593900.0594940.0595990.059704
0.059809
0.0598090.0599140.0600190.0601240.060230
0.0603350.0604410.0605470.0606530.060759
0.0608660.0609720.0610790.0611860.061292
0.0614000.0615070.0616140.0617210.061829
0.0619370.0620450.0621530.0622610.062369
0.0624780.0625860.0626950.0628040.062913
0.0630220.0631310.0632410.0633500.063460
0.0635700.0636800.0637900.0639010.064011
0.0641220.0642320.0643430.0644540.064565
0.0646770.0647880.0649000.0650120.065123
0.0652360.0653480.0654600.0655730.065685
0.0657980.0659110.0660240.0661370.066250
0.066364
23° 24° 25° 26° 27° 28° 29° 30° 31°
Numerical Formulas and Tables
0.0734490.0735720.0736950.0738180.073941
0.0740640.0741880.0743110.0744350.074559
0.0746840.0748080.0749320.0750570.075182
0.0753070.0754320.0755570.0756830.075808
0.0759340.0760600.0761860.0763120.076439
0.0765650.0766920.0768190.0769460.077073
0.0772000.0773280.0774550.0775830.077711
0.0778390.0779680.0780960.0782250.078354
0.0784830.0786120.0787410.0788710.079000
0.0791300.0792600.0793900.0795200.079651
0.0797810.0799120.0800430.0801740.080306
0.0804370.0805690.0807000.0808320.080964
0.081097
0.0663640.0664780.0665910.0667050.066819
0.0669340.0670480.0671630.0672770.067392
0.0675070.0676220.0677380.0678530.067969
0.0680840.0682000.0683160.0684320.068549
0.0686650.0687820.0688990.0690160.069133
0.0692500.0693670.0694850.0696020.069720
0.0698380.0699560.0700750.0701930.070312
0.0704300.0705490.0706680.0707880.070907
0.0710260.0711460.0712660.0713860.071506
0.0716260.0717470.0718670.0719880.072109
0.0722300.0723510.0724730.0725940.072716
0.0728380.0729590.0730820.0732040.073326
0.073449
20
inv α = tan α − αInvolute Function Table
0001020304
0506070809
1011121314
1516171819
2021222324
2526272829
3031323334
3536373839
4041424344
4546474849
5051525354
5556575859
60
32°
0.0810970.0812290.0813620.0814940.081627
0.0817600.0818940.0820270.0821610.082294
0.0824280.0825620.0826970.0828310.082966
0.0831000.0832350.0833710.0835060.083641
0.0837770.0839130.0840490.0841850.084321
0.0844580.0845940.0847310.0848680.085005
0.0851420.0852800.0854180.0855550.085693
0.0858320.0859700.0861080.0832470.086386
0.0865250.0866640.0868040.0869430.087083
0.0872230.0873630.0875030.0876440.087784
0.0879250.0880660.0882070.0883480.088490
0.0886310.0887730.0889150.0890570.089200
0.089342
0.0893420.0894850.0896280.0897710.089914
0.0900580.0902010.0903450.0904890.090633
0.0907770.0909220.0910670.0912110.091356
0.0915020.0916470.0917920.0919380.092084
0.0922300.0923770.0925230.0926700.092816
0.0929630.0931110.0932580.0934060.093533
0.0937010.0938490.0939980.0941460.094295
0.0944430.0945920.0947420.0948910.095041
0.0951900.0953400.0954900.0956410.095791
0.0959420.0960930.0962440.0963950.096546
0.0966980.0968500.0970020.0971540.097306
0.0974590.0976110.0977640.0979170.098071
0.098224
0.0982240.0983780.0985310.0986850.098840
0.0989940.0991490.0993030.0994580.099614
0.0997690.0999240.1000800.1002360.100392
0.1005480.1007050.1008620.1010190.101176
0.1013330.1014900.1016480.1018060.101964
0.1021220.1022800.1024390.1025980.102757
0.1029160.1030750.1032350.1033950.103555
0.1037150.1038750.1040360.1041960.104357
0.1045180.1046800.1048410.1050030.105165
0.1053270.1054890.1056520.1058140.105977
0.1061400.1063040.1064670.1066310.106795
0.1069590.1071230.1072880.1074520.107617
0.107782
0.1077820.1079480.1081130.1082790.108445
0.1086110.1087770.1089430.1091100.109277
0.1094440.1096110.1097790.1099470.110114
0.1102830.1104510.1106190.1107880.110957
0.1111260.1112950.1114650.1116350.111805
0.1119750.1121450.1123160.1124860.112657
0.1128290.1130000.1131710.1133430.113515
0.1136870.1138600.1140320.1142050.114378
0.1145520.1147250.1148990.1150730.115247
0.1154210.1155950.1157700.1159450.116120
0.1162960.1164710.1166470.1168230.116999
0.1171750.1173520.1175290.1177060.117883
0.118061
0.1180610.1182380.1184160.1185940.118772
0.1189510.1191300.1193090.1194880.119667
0.1198470.1200270.1202070.1203870.120567
0.1207480.1209290.1211100.1212910.121473
0.1216550.1218370.1220190.1222010.122384
0.1225670.1227500.1229330.1231160.123300
0.1234840.1236680.1238530.1240370.124222
0.1244070.1245920.1247780.1249640.125150
0.1253360.1255220.1257090.1258950.126083
0.1262700.1264570.1266450.1268330.127021
0.1272090.1273980.1275870.1277760.127965
0.1281550.1283440.1285340.1287250.128915
0.129106
0.1291060.1292960.1294880.1296790.129870
0.1300620.1302540.1304460.1306390.130832
0.1310250.1312180.1314110.1316050.131798
0.1319930.1321870.1323810.1325760.132771
0.1329660.1331620.1333570.1335530.133750
0.1339460.1341430.1343390.1345360.134734
0.1349310.1351290.1353270.1355250.135724
0.1359230.1361220.1363210.1365200.136720
0.1369200.1371200.1373200.1375210.137722
0.1379230.1381240.1383260.1385280.138730
0.1389320.1391340.1393370.1395400.139743
0.1399470.1401510.1403550.1405590.140763
0.140968
0.1409680.1411730.1413780.1415830.141789
0.1419950.1422010.1424080.1426140.142821
0.1430280.1432360.1434430.1436510.143859
0.1440670.1442760.1444850.1446940.144903
0.1451130.1453230.1455330.1457430.145954
0.1461650.1463760.1465870.1467980.147010
0.1472220.1474350.1476470.1478600.148073
0.1482860.1485000.1487140.1489280.149142
0.1493570.1495720.1497870.1500020.150218
0.1504330.1506500.1508660.1510830.151299
0.1515160.1517340.1519510.1521690.152388
0.1526060.1528250.1530430.1532630.153482
0.153702
0.153700.153920.154140.154360.15458
0.154800.155030.155250.15547015569
0.155910.156140.156360.156580.15680
0.157030.157250.157480.157700.15793
0.158150.158380.158600.158830.15905
0.159280.159510.159730.159960.16019
0.160410.160640.160870.161100.16133
0.161560.161780.162010.162240.16247
0.162700.162930.163170.163400.16363
0.163860.164090.164320.164560.16479
0.165020.166250.165490.165720.16596
0.166190.166420.166660.166890.16713
0.16737
33° 34° 35° 36° 37° 38° 39° 40° 41°
Numerical Formulas and Tables
0.167370.167600.167840.168070.16831
0.168550.168790.169020.169260.16950
0.169740.169980.170220.170450.17069
0.170930.171170.171420.171660.17190
0.172140.172380.172620.172860.17311
0.173350.173590.173830.174080.17432
0.174570.174810.175060.175300.17555
0.175790.176040.176280.176530.17678
0.177020.177270.177520.177770.17801
0.178260.178510.178760.179010.17926
0.179510.179760.180010.180260.18051
0.180760.181010.181270.181520.18177
0.18202
21
inv α = tan α − αInvolute Function Table
0001020304
0506070809
1011121314
1516171819
2021222324
2526272829
3031323334
3536373839
4041424344
4546474849
5051525354
5556575859
60
42°
0.182020.182280.182530.182780.18304
0.183290.183550.183800.184060.18431
0.184570.184820.185080.185340.18559
0.185850.186110.186370.186620.18688
0.187140.187400.187660.187920.18818
0.188440.188700.188960.189220.18948
0.189750.190010.190270.190530.19080
0.191060.191320.191590.191850.19212
0.192380.192650.192910.193180.19344
0.193710.193980.194240.194510.19478
0.195050.195320.195580.195850.19612
0.196390.196660.196930.197200.19747
0.19774
0.197740.198020.198290.198560.19883
0.199100.199380.199650.199920.20020
0.200470.200750.201020.201300.20157
0.201850.202120.202400.202680.20296
0.203230.203510.203790.204070.20435
0.204630.204900.205180.205460.20575
0.206030.206310.206590.206870.20715
0.207430.207720.208000.208280.20857
0.208850.209140.209420.209710.20999
0.210280.210560.210850.211140.21142
0.211710.212000.212290.212570.21286
0.213150.213440.213730.214020.21431
0.21460
0.214600.214890.215180.215480.21577
0.216060.216350.216650.216940.21723
0.217530.217820.218120.218410.21871
0.219000.219300.219600.219890.22019
0.220490.220790.221080.221380.22168
0.221980.222280.222580.222880.22318
0.223480.223780.224090.224390.22469
0.224990.225300.225600.225900.22621
0.226510.226820.227120.227430.22773
0.228040.228350.228650.228960.22927
0.229580.229890.230200.230500.23081
0.231120.231430.231740.232060.23237
0.23268
0.232680.232990.233300.233620.23393
0.234240.234560.234870.235190.23550
0.235820.236130.236450.236760.23708
0.237400.237720.238030.238350.23867
0.238990.239310.239630.239950.24027
0.240590.240910.241230.241560.24188
0.242200.242530.242850.243170.24350
0.243820.244150.244470.244800.24512
0.245450.245780.246110.246430.24676
0.247090.247420.247750.248080.24841
0.248740.249070.249400.249730.25006
0.250400.250730.251060.251400.25173
0.25206
0.252060.252400.252730.253070.25341
0.253740.254080.254420.254750.25509
0.255430.255770.256110.256450.25679
0.257130.257470.257810.258150.25849
0.258830.259180.259520.259860.26021
0.260550.260890.261240.261590.26193
0.262280.262620.262970.263320.26367
0.264010.264360.264710.265060.26541
0.265760.266110.266460.266820.26717
0.267520.267870.268230.268580.26893
0.269290.269640.270000.270350.27071
0.271070.271420.271780.272140.27250
0.27285
0.272850.273210.273570.273930.27429
0.274650.275010.275380.275740.27610
0.276460.276830.277190.277550.27792
0.278280.278650.279020.279380.27975
0.280120.280480.280850.281220.28159
0.281960.282330.282700.283070.28344
0.283810.284180.284550.284930.28530
0.285670.286050.286420.286800.28717
0.287550.287920.288300.288680.28906
0.289430.289810.290190.290570.29095
0.291330.291710.292090.292470.29286
0.293240.293620.294000.294390.29477
0.29516
0.295160.295540.295930.296310.29670
0.297090.297470.297860.298250.29864
0.299030.299420.299810.300200.30059
0.300980.301370.301770.302160.30255
0.302950.303340.303740.304130.30453
0.304920.305320.305720.306110.30651
0.306910.307310.307710.308110.30851
0.308910.309310.309710.310120.31052
0.310920.311330.311730.312140.31254
0.312950.313350.313760.314170.31457
0.314980.315390.315800.316210.31662
0.317030.317440.317850.318260.31868
0.31909
0.319090.319500.319920.320330.32075
0.321160.321580.321990.322410.32283
0.323240.323660.324080.324500.32492
0.325340.325760.326180.326610.32703
0.327450.327870.328300.328720.32915
0.329570.330000.330420.330850.33128
0.331710.332130.332560.332990.33342
0.333850.334280.334710.335150.33558
0.336010.336450.336880.337310.33775
0.338180.338620.339060.339490.33993
0.340370.340810.341250.341690.34213
0.342570.343010.343450.343890.34434
0.34478
0.344780.345220.345670.346110.34656
0.347000.347450.347900.348340.34879
0.349240.349690.350140.350590.35104
0.351490.351940.352400.352850.35330
0.353760.354210.354670.355120.35558
0.356040.356490.356950.357410.35787
0.358330.358790.359250.359710.36017
0.360630.361100.361560.362020.36249
0.362950.363420.363880.364350.36482
0.365290.365750.366220.366690.36716
0.367630.368100.368580.369050.36952
0.369990.370470.370940.371420.37189
0.37237
43° 44° 45° 46° 47° 48° 49° 50° 51°
Numerical Formulas and Tables
0.433900.434460.435010.435560.43611
0.436670.437220.437780.438330.43889
0.439450.440010.440570.441130.44169
0.442250.442810.443370.443930.44450
0.445060.445630.446190.446760.44733
0.447890.448460.449030.449600.45017
0.450740.451320.451890.452460.45304
0.453610.454190.454760.455340.45592
0.456500.457080.457660.458240.45882
0.459400.459980.460570.461150.46173
0.462320.462910.463490.464080.46467
0.465260.465850.466440.467030.46762
0.46822
0.372370.372850.373320.373800.37428
0.374760.375240.375720.376200.37668
0.377160.377650.378130.378610.37910
0.379580.380070.380550.381040.38153
0.382020.382510.382990.383480.38397
0.384460.384960.385450.385940.38643
0.386930.387420.387920.388410.38891
0.389410.389900.390400.390900.39140
0.391900.392400.392900.393400.39390
0.394410.394910.395410.395920.39642
0.396930.397430.397940.398450.39896
0.399470.399980.400490.401000.40151
0.40202
22
inv α = tan α − αInvolute Function Table
0001020304
0506070809
1011121314
1516171819
2021222324
2526272829
3031323334
3536373839
4041424344
4546474849
5051525354
5556575859
60
52°
0.402020.402530.403050.403560.40407
0.404590.405110.405620.406140.40666
0.407170.407690.408210.408730.40925
0.409770.410300.410820.411340.41187
0.412390.412920.413440.413970.41450
0.415020.415550.416080.416610.41714
0.417670.418200.418740.419270.41980
0.420340.420870.421410.421940.42248
0.423020.423550.424090.424630.42517
0.425710.426250.426800.427340.42788
0.428430.428970.429520.430060.43061
0.431160.431710.432250.432800.43335
0.43390
0.468220.468810.469400.470000.47060
0.471190.471790.472390.472990.47359
0.474190.474790.475390.475990.47660
0.477200.477800.478410.479020.47962
0.480230.480840.481450.482060.48267
0.483280.483890.484510.485120.48574
0.486350.486970.487580.488200.48882
0.489440.490060.490680.491300.49193
0.492550.493170.493800.494420.49505
0.495680.496300.496930.497560.49819
0.498820.499450.500090.500720.50135
0.501990.502630.503260.503900.50454
0.50518
0.505180.505820.506460.507100.50774
0.508380.509030.509670.510320.51096
0.511610.512260.512910.513560.51421
0.514860.515510.516160.516820.51747
0.518130.518780.519440.520100.52076
0.521410.522070.522740.523400.52406
0.524720.525390.526050.526720.52739
0.528050.528720.529390.530060.53073
0.531410.532080.532750.533430.53410
0.534780.535460.536130.536810.53749
0.538170.538850.539540.540220.54090
0.541590.542280.542960.543650.54434
0.54503
0.545030.545720.546410.547100.54779
0.548490.549180.549880.550570.55127
0.551970.552670.553370.554070.55477
0.555470.556180.556880.557590.55829
0.559000.559710.560420.561130.56184
0.562550.563260.563980.564690.56541
0.566120.566840.567560.568280.56900
0.569720.570440.571160.571880.57261
0.573330.574060.574790.575520.57625
0.576980.577710.578440.579170.57991
0.580640.581380.582110.582850.58359
0.584330.585070.585810.586560.58730
0.58804
0.588040.588790.589540.590280.59103
0.591780.592530.593280.594030.59479
0.595540.596300.597050.597810.59857
0.599330.600090.600850.601610.60237
0.603140.603900.604670.605440.60620
0.606970.607740.608510.609290.61006
0.610830.611610.612390.613160.61394
0.614720.615500.616280.617060.61785
0.618630.619420.620200.620990.62178
0.622570.623360.624150.624940.62574
0.626530.627330.628120.628920.62972
0.630520.631320.632120.632930.63373
0.63454
0.634540.635340.636150.636960.63777
0.638580.639390.640200.641020.64183
0.642650.643460.644280.645100.64592
0.646740.647560.648390.649210.65004
0.650860.651690.652520.653350.65418
0.655010.655850.656680.657520.65835
0.659190.660030.660870.661710.66255
0.663400.664240.665090.665940.66678
0.667630.668480.669330.670190.67104
0.671890.672750.673610.674470.67532
0.676180.677050.677910.678770.67964
0.680500.681370.682240.683110.68398
0.68485
0.684850.685730.686600.687480.68835
0.689230.690110.690990.691870.69276
0.693640.694520.695410.696300.69719
0.698080.698970.699860.700750.70165
0.702540.703440.704340.705240.70614
0.707040.707940.708850.709750.71066
0.711570.712480.713390.714300.71521
0.716130.717040.717960.718880.71980
0.720720.721640.722560.723490.72441
0.725340.726270.727200.728130.72906
0.729990.730930.731860.732800.73374
0.734680.735620.736560.737510.73845
0.73940
0.739400.740340.741290.742240.74319
0.744150.745100.746060.747010.74797
0.748930.749890.750850.751810.75278
0.753750.754710.755680.756650.75762
0.758590.759570.760540.761520.76250
0.763480.764460.765440.766420.76741
0.768390.769380.770370.771360.77235
0.773340.774340.775330.776330.77733
0.778330.779330.780330.781340.78234
0.783350.784360.785370.786380.78739
0.788400.789420.790440.791460.79247
0.793500.794520.795540.796570.79759
0.79862
53° 54° 55° 56° 57° 58° 59° 60° 61°
Numerical Formulas and Tables
0.798620.799650.800680.801720.80275
0.803780.804820.805860.806900.80794
0.808980.810030.811070.812120.81317
0.814220.815270.816320.817380.81844
0.819490.820550.821610.822670.82374
0.824800.825870.826940.828010.82908
0.830150.831230.832300.833380.83446
0.835540.836620.837700.838790.83987
0.840960.842050.843140.844240.84533
0.846430.847520.848620.849720.85082
0.851930.853030.854140.855250.85636
0.857470.858580.859700.860820.86193
0.86305
23
inv α = tan α − αInvolute Function Table
0001020304
0506070809
1011121314
1516171819
2021222324
2526272829
3031323334
3536373839
4041424344
4546474849
5051525354
5556575859
60
62°
0.863050.864170.865300.866420.86755
0.868680.869800.870940.872070.87320
0.874340.875480.876620.877760.87890
0.880040.881190.882340.883490.88464
0.885790.886940.888100.889260.89042
0.891580.892740.893900.895070.89624
0.897410.898580.899750.900920.90210
0.903280.904460.905640.906820.90801
0.909190.910380.911570.912760.91396
0.915150.916350.917550.918750.91995
0.921150.922360.923570.924780.92599
0.927200.928420.929630.930850.93207
0.93329
0.933290.934520.935740.936970.93820
0.939430.940660.941900.943130.94437
0.945610.946850.948100.949340.95059
0.951840.953090.954340.955600.95686
0.958120.959380.960640.961900.96317
0.964440.965710.966980.968250.96953
0.970810.972090.973370.974650.97594
0.977220.978510.979800.981100.98239
0.983690.984990.986290.987590.98890
0.990200.991510.992820.994140.99545
0.996770.998080.999411.000731.00205
1.003381.004711.006041.007371.00871
1.01004
1.010041.011381.012721.014071.01541
1.016761.018111.019461.020811.02217
1.023521.024881.026241.027611.02897
1.030341.031711.033081.034461.03583
1.037211.038591.039971.041361.04274
1.044131.045521.046921.048311.04971
1.051111.052511.053911.055321.05673
1.058141.059551.060971.062381.06380
1.065221.066651.068071.069501.07093
1.072361.073801.075241.076671.07812
1.079561.081001.082451.083901.08536
1.086811.088271.089731.091191.09265
1.09412
1.094121.095591.097061.098531.10001
1.101491.102971.104451.105931.10742
1.108911.110401.111901.113391.11489
1.116391.117901.119401.120911.12242
1.123931.125451.126971.128491.13001
1.131541.133061.134591.136131.13766
1.139201.140741.142281.143831.14537
1.146921.148471.150031.151591.15315
1.154711.156271.157841.159411.16098
1.162561.164131.165711.167291.16888
1.170471.172061.173651.175241.17684
1.178441.180041.181651.183261.18487
1.18648
1.186481.188101.189721.191341.19296
1.194591.196221.197851.199481.20112
1.202761.204401.206041.207691.20934
1.211001.212651.214311.215971.21763
1.219301.220971.222641.224321.22599
1.227671.229361.231041.232731.23442
1.236121.237811.239511.241221.24292
1.244631.246341.248051.249771.25149
1.253211.254941.256661.258391.26013
1.261871.263601.264351.267091.26884
1.270591.272351.274101.275861.27762
1.279361.281161.282931.284701.28648
1.28826
1.288261.290051.291831.293621.29541
1.297211.299011.300811.302621.30442
1.306231.308051.309861.311681.31351
1.315331.317161.318991.320831.32267
1.324511.326351.328201.330051.33191
1.333761.335621.337491.339351.34122
1.343101.344971.346851.348741.35062
1.352511.354401.356301.358201.36010
1.362011.363911.365831.367741.36966
1.371581.373511.375441.377371.37930
1.381241.383181.385131.387081.38903
1.390981.392941.394901.396871.39884
1.40081
1.400811.402791.404771.406751.40874
1.410731.412721.414721.416721.41872
1.420731.422741.424751.426771.42879
1.430811.432841.434871.436911.43895
1.440991.443041.445091.447141.44920
1.451261.453321.455391.457461.45954
1.461621.463701.465791.467881.46997
1.472071.474171.476271.478381.48050
1.482611.484731.486861.488981.49112
1.493251.495391.497531.499681.50183
1.503991.506141.508311.510471.51264
1.514821.517001.519181.521361.52355
1.52575
1.525751.527941.530151.532351.53456
1.536781.538991.541221.543441.54567
1.547911.550141.552391.554631.55688
1.559141.561401.563661.565931.56820
1.570471.572751.575031.577321.57961
1.581911.584211.586521.588821.59114
1.593461.595781.598101.600431.60277
1.605111.607451.609801.612151.61451
1.616871.619231.621601.623981.62636
1.628741.631131.633521.635921.63832
1.640721.643131.645551.647971.65039
1.652821.655251.657691.660131.66258
1.66503
1.665031.667481.669941.672411.67488
1.677351.679831.682321.684801.68730
1.689801.692301.694811.697321.69984
1.702361.704881.707421.709951.71249
1.715041.717591.720151.722711.72527
1.727851.730421.733001.735591.73818
1.740771.743381.745981.748591.75121
1.753831.756461.759091.761721.76436
1.767011.769661.772321.774981.77765
1.780321.783001.785681.788371.79106
1.793761.796471.799181.801891.80461
1.807341.810071.812801.815551.81829
1.82105
63° 64° 65° 66° 67° 68° 69° 70° 71°
Numerical Formulas and Tables
2.684332.689022.693712.698422.70314
2.707872.712622.717372.722142.72692
2.731712.736512.741332.746162.75100
2.755852.760712.765592.770482.77538
2.780292.785222.790162.795112.80007
2.805052.810042.815042.820062.82508
2.830122.835182.840242.845322.85041
2.855522.860642.865772.870922.87607
2.881252.886432.891632.896842.90207
2.907312.912562.917832.923112.92840
2.933712.939032.944372.949722.95509
2.960462.965862.971262.976692.98212
2.98757
1.821051.823801.826571.829341.83211
1.834891.837681.840471.843261.84607
1.848881.851691.854511.857331.86016
1.863001.865841.868691.871541.87440
1.877261.880141.883011.885891.88878
1.891671.894571.897481.900391.90331
1.906231.909161.912101.915041.91798
1.920941.923891.926861.929831.93281
1.935791.938781.941781.944781.94779
1.950801.953821.956851.959881.96292
1.965961.969011.972071.975141.97821
1.981281.984371.987461.990551.99365
1.99676
24
inv α = tan α − αInvolute Function Table
0001020304
0506070809
1011121314
1516171819
2021222324
2526272829
3031323334
3536373839
4041424344
4546474849
5051525354
5556575859
60
72°
1.996761.999882.003002.006132.00926
2.012402.015552.018712.021872.02504
2.028212.031392.034582.037772.04097
2.044182.047402.050622.053851.05708
2.060322.063572.066832.070092.07336
2.076642.079922.083212.086512.08981
2.093132.096452.099772.103102.10644
2.109792.113152.116512.119882.12325
2.126642.130032.133432.136832.14024
2.143662.147092.150532.153972.15742
2.160882.164342.167812.171302.17478
2.178282.181782.185292.188812.19234
2.19587
2.195872.199412.202962.206522.21008
2.213662.217242.220832.334422.22803
2.231642.235262.238892.242532.24617
2.249832.253492.257162.260832.26452
2.268212.271922.275632.279352.28307
2.286812.290552.294302.298072.30184
2.305612.309402.313192.317002.32081
2.324632.328462.332302.336152.34000
2.343872.347742.351622.355512.35941
2.363322.367242.371172.375112.37905
2.383002.386972.390942.394922.39891
2.402912.406922.411942.414972.41901
2.42305
2.423052.427112.431182.435252.43934
2.443432.447532.451652.455772.45990
2.464052.468202.472362.476532.48071
2.484912.489112.493322.497542.50177
2.506012.510272.514532.518802.52308
2.527372.531682.535992.540312.54465
2.548992.553342.557712.562082.56647
2.570872.575272.579692.584122.58856
2.593012.597472.601942.606422.61092
2.615422.619942.624462.629002.63355
2.638112.642682.647262.651862.65646
2.661082.665712.670342.675002.67966
2.68433
2.987572.993042.998523.004013.00952
3.015043.020583.026133.031703.03728
3.042883.048493.054123.059773.06542
3.071103.076793.082493.088213.09395
3.099703.105463.111253.117043.12286
3.128693.134533.140403.146273.15217
3.158083.164013.169953.175913.18188
3.187883.193893.199913.205953.21201
3.218093.224183.230293.236423.24257
3.248733.254913.261103.267323.27355
3.279803.286063.292353.298653.30497
3.311313.317673.324043.330433.33684
3.34327
3.343273.349723.356193.362673.36918
3.375703.382243.388803.395383.40197
3.408593.415233.421883.428563.43525
3.441973.448703.455453.462223.46902
3.475833.482663.489523.496393.50328
3.510203.517133.524083.531063.53806
3.545073.552113.559173.566253.57335
3.580473.587623.594783.601973.60918
3.616413.623663.630943.638233.64555
3.652893.660263.667643.675053.68248
3.689933.697413.704913.712433.71998
3.727553.735143.742753.750393.75806
3.76574
3.765743.773453.781193.788953.79673
3.804543.812373.830233.828113.83601
3.843953.851903.859883.867893.87592
3.883983.892063.900173.908303.91646
3.924653.932863.941103.949373.95766
3.965983.974333.982703.991103.99953
4.007984.016464.024974.033514.04207
4.050674.059294.067944.076624.08532
4.094064.102824.111624.120444.12929
4.138174.147084.156021.164994.17399
4.183024.192084.201184.210304.21945
4.228634.237854.247094.256374.26568
4.27502
4.275024.284394.293794.303234.31270
4.322204.331734.341304.350904.36053
4.370204.379904.389634.399404.40920
4.419034.428904.438804.448744.45871
4.468724.478774.488854.498964.50911
4.519304.529524.539784.550074.56041
4.570774.581184.591624.602104.61262
4.623184.633774.644414.655084.66579
4.676544.687334.698164.709024.71993
4.730884.741864.752894.763964.77507
4.786224.797414.808654.819924.83124
4.842604.854004.865444.876934.88846
4.90003
4.900034.911654.923314.935024.94677
4.958564.970404.982294.994225.00620
5.018225.030295.042405.054565.06677
5.079025.091335.103685.116085.12852
5.141025.153565.166165.178805.19149
5.204245.217035.229875.242775.25572
5.268715.281765.294865.308025.32122
5.334485.347805.361175.374595.38806
5.401595.415185.428825.442515.45626
5.470075.483945.497865.511845.52588
5.539975.554135.568345.582615.59694
5.611335.625785.640305.654875.66950
5.68420
73° 74° 75° 76° 77° 78° 79° 80° 81°
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