Area Volume Calculation in Structures

8/19/2019 Area Volume Calculation in Structures

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Contents1.0 Measurements for Regular Areas..........................................................................2

1.1 Lines and Angles....................................................................................... ........ 2

1.2 Triangles............................................................................................................ 3

1.2.1 Perimeter of a Triangle........................................................................ ........ 3

1.2.2 Pythagorean’s Theorem................................................................. ............. 3

1.2.3 Area of a Triangle................................................................................. ....... 3

1.3 Quadrilaterals....................................................................................................4

1.3.1 Area and Perimeter of Quadrilaterals..................................................... .....4

1.4 ir!les........................................................................................... .............. ...... 4

1.4.1 Perimeter and Area of a ir!le.....................................................................4

2.0 Measurements for "rregular Areas................................................................ ........ 4

2.1 The Tra#e$oidal Rule......................................................................... ............... .%

2.2 The &im#son’s Rule.................................................................................... ....... %

2.3 Areas 'y "ntegration.......................................................................................... %

3.0 Measurements for al!ulating (olume.......................................................... .......%

3.1 (olumes 'y "ntegration...................................................................... .............. . )

3.1.1 (olume of a ur*e....................................................................................... )

3.1.2 (olume of a &hell............................................................................ ............ +4.0 Trigonometry to &ol*e for antile*ers....................................................... ........+

%.0 entroids........................................................................................................... ,

%.0 -imensionless rou#s....................................................................................... ,

%.1 Reynold’s /um'er.......................................................................................... ,

).0 nits and nit on*ersions......................................................................... ...... ,

1.0 Measurements for Regular Areas

1.1 Lines and Anglesith res#e!t to geometry #oints lines and #lanes are the 'asis for all

!al!ulations. "t is im#ortant to note that these are relati*e terms they are a!!e#ted


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ithout 'eing de5ned and are the foundation for more rele*ant geometri!!al!ulations.

To understand the !on!e#t of an angle one must 5rst 'e familiar ith thea'o*e noted !om#onents of a line. A #oint is nothing more than a lo!ation in s#a!e.

A line !onsists of a !olle!tion of #oints along a strait #ath hi!h is !ontinuous iseither dire!tion. "t is im#ortant to note that a line has not end#oints. A line segmentis a #ortion of a line that has t o end #oints. A ray is a line ith one end hi!h is!ontinuous in one dire!tion. The 56ed end #oint of a ray is the *erte6 of an angle.

The rotation a'out this #oint is hat forms an angle. A full rotation a'out a *erte6 is3)07. 8ther im#ortant angles are9

o Right angle9 ,07o &trait angle9 1:07o A!ute angle9 ; ,07o 8'tuse angle9 < ,07

"f t o lines interse!t at right angles or at ,07 to one another it !an 'e saidthat these line are #er#endi!ular. 8##ositely if t o lines tra*el alone the same#lane 'ut ne*er interse!t !an 'e said to 'e #arallel. "dentifying these line ty#es isim#ortant to !on!e#ts dis!ussed later in this re#ort.

hen an angle is !reated at a #oint on a line it !reates a #air of angles one'eing o'tuse and the other 'eing a!ute. "f the sum of these t o angles is 1:07 the#air of angles is said to 'e su##lementary. &imilar to this !on!e#t if t o angles #airtogether and ha*e a sum ,07 the #aired angles are said to 'e !om#lementary.

"n addition to su##lementary and !om#lementary angles there are a grou# ofs#e!ial #aired angles these angles are =no n as ad>a!ent and *erti!al angles. T oangles hi!h share the same *erte6 and ha*e one side in !ommon are said to 'ead>a!ent. T o angles hi!h share a *erte6 and are o##osite to one another are saidto 'e *erti!al angles it is im#ortant to note that *erti!al angles are e?ual to oneanother.

The a'ility to identify these angles and angle #airings is fundamental tosol*ing #ro'lems asso!iated ith trigonometry. The 5nal !on!e#t dis!ussed in thisse!tion ill 'e the notion of a trans*ersal. A trans*ersal is a line hi!h interse!tst o or more other lines #arallel or not to form a grou# of angles. hen a

trans*ersal !rosses t o or more #arallel lines !orres#onding interior angles and!orres#onding e6terior angles are !reated. These !orres#onding o##osite angles arethen e?ual to one another mu!h the same ay *erti!al angles are e?ual to oneanother.


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1.2 Triangles The triangle is ?uite #ossi'le the

most im#ortant and most hea*ilystudied sha#e in geometry. A triangle isa !losed sha#e ith three sides hi!h

for three angles that ha*e the sum of 1:07.There are three ty#es of triangles9s!alene isos!eles and e?uilateral. As!alene triangle has no t o sides of e?ual length. An isos!eles triangle hast o sides of e?ual length and the t o angles !reated at the 'ase of the triangle aree?ual to one another. An e?uilateral triangle has all three sides of e?ual length andall of its interior angles are e?ual to )07. "f any t o angles of a triangle are =no thethird !an 'e found 'y su'tra!ting the sum of the t o =no n angles from 1:07.

There is ho e*er a fourth ty#e of triangle the right triangle and it is themost useful in geometry. A right triangle has one angle of ,07. The side of thetriangle o##osite to the ,07 angle is !alled the hy#otenuse and the other t o sidesare referred to as legs.

1.2.1 Perimeter of a Triangle The #erimeter of any sha#e is the sum of the lengths of all of its sides. "f the

three sides of a triangle !an 'e !onsidered as side A side @ and side then the

formula for the #erimeter of any triangle !an 'e e6#ressed as9 P= A+B+C .

1.2.2 Pythagorean’s TheoremAs stated earlier in this re#ort the most im#ortant and fundament triangle in

geometry is the right triangle. 8ne of the most #o erful tools e ha*e hen it!omes to !al!ulations asso!iated ith the right triangle is Pythagorean’s Theorem.

The theorem states that in right triangles the s?uare of the length of thehy#otenuse e?uals the sum of the s?uares of the lengths of the other t o sides.B "nlayman’s terms this means if any t o sides of a right triangle are =no n then thelength of the third un=no n side !an 'e deri*ed. The formula is as follo s9

A2 +B2 = C 2 here is the hy#otenuse.

1.2.3 Area of a TriangleArea !an 'e de5ned as gi*ing measure to the surfa!e of a 5gure.B The

formula for !al!ulating the area of ?uadrilaterals !losed sha#es ith four sidessu!h as s?uares and re!tangles is the length of its 'ase multi#lied 'y its height or is

often thought of as its length times its idth and is e6#ressed as9 A= B∗ H or

A= L∗W . A triangle !an 'e thought of as half of one of these ?uadrilaterals so

Figure 1 our!e"math#orld.#olfram.!om


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the formula for !al!ulating the area of a triangle is e6#ressed as9 A=1

2B∗ H . Any

sha#e of triangle ith the same 'ase and height ill therefore ha*e the same area.

Another #rin!i#le e?uation for !al!ulating the area of a triangle is the Cero’sformula. The Cero’s formula !an only 'e a##lied hen the length of all three sidesof the triangle is =no n and the triangle !ontains no right angles. The Cero’s

formula is e6#ressed as9 A= √ s(s− a )(s− b)(s− c) here a ' and ! are the sides

of the triangle and s=1

2(a +b+c) .

1.3 $uadrilaterals@rieDy noted a'o*e ?uadrilaterals are !losed sha#es ith four sides that

form four interior angles. There are three ty#es of ?uadrilaterals9 the #arallelogramthe re!tangle and the tra#e$oid. A #arallelogram is a sha#e here o##osite sidesare e?ual in length #arallel to one another and do not form any right angles. Are!tangle is sim#ly a #arallelogram here interse!ting sides meet at right angles orare #er#endi!ular to one another. A re!tangle hose sides are all of e?ual length is=no n as a s?uare. A tra#e$oid is a ?uadrilateral that only has t o sides hi!h are#arallel to one another.

1.3.1 Area and Perimeter of $uadrilateralsEor any sha#e #erimeter !al!ulations are al ays the same the sum of the

length of all its side. This !al!ulation has 'een #re*iously dis!ussed for triangle and

ill not 'e e6#lored in this se!tion.

There is a set of four e?uations for 5nding the area of any ?uadrilateral9

• A= s 2 for s?uares here s is the length of any side• A= l∗w for re!tangles here l is the length and is the idth• A= b∗h for #arallelograms here ' is the 'ase and h is the height

• A=

1

2h (b1 +b 2 ) for tra#e$oids here b1 and b2 are the 'ases Fto# and

'ottomG and h is height

1.% Cir!les The 5nal sha#ed dis!ussed in this se!tion of the re#ort ill 'e !ir!les. @efore

going into the !al!ulations asso!iated ith !ir!les there are a fe im#ortant thingsto note a'out the sha#e. Eirstly all #oints of a !ir!le are e?ual distan!e from its!enter. The distan!e from the !ir!le’s !enter to a #oint on the !ir!le is referred to as


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the radius. The distan!e greatest distan!e from #oint to #oint that !rosses the !ir!le!enter #oint is =no n as the diameter. The diameter !an 'e thought of as t i!e the

radius d= 2 r .

1.%.1 Perimeter and Area of a Cir!le There are many !om#le6ities and !al!ulations asso!iated ith !ir!les this

re#ort ill refrain from those !om#le6ities and fo!us only on #erimeter and area!al!ulations. The #erimeter of a !ir!le is =no n as the !ir!umferen!e. "f the radius of a !ir!le is =no n then the !ir!umferen!e !an 'e

!al!ulated using the formula c= 2 πr or c= πd

here r is the radius and d is the diameter. Thearea of a !ir!le is !an 'e !al!ulated using the same

!om#onent using the formula A= π r2

or

A= π d2

2 again here r is the radius and d is the

diameter.

2.0 Measurements for &rregular Areas Thus far in this re#ort all sha#es

dis!usses ha*e 'een asso!iated ith#re!ise formulas for !al!ulating the sha#e’se6a!t area. "n real orld a##li!ations of geometry the 'oundaries of areas tend not to 'e regular 'ut irregular and for this

e ha*e t o methods for a##ro6imating area the tra#e$oidal rule and the &im#sonrule. This re#ort ill also dis!uss !al!ulating areas using integration.

2.1 The Tra'e(oidal Rule To utili$e the tra#e$oidal rule the irregular area must 5rst 'e di*ided into

tra#e$oids ea!h of hi!h ill ha*e e?ual height. This !an 'e a!hie*ed 'y dra ing#arallel lines at n distan!e a#art 'et een the 'ounds of sha#es area. The sum of allthe areas of these tra#e$oids ill gi*e you an a##ro6imation of the sha#es totalarea.

"f e !onsider the length of ea!h #arallel line dra n to 'e y1 , y2 , y3 … y4

res#e!ti*ely and e !onsider the distan!e 'et een the #arallel lines to 'e h thenthe e?uation deri*ed for the sha#es area is9

Figure 2 our!e" )ashington '. *+


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A ≅ h2 ( y0 + y1 )+

h2 ( y1 + y2 )+

h2

( y2 + y3 )…+h2 ( yn− 2 + yn− 1 )+h2

( yn− 1 + yn) hi!h sim#li5es to9

A ≅ h2

( y0 +2 y1 +2 y2 …+2 yn− 1 + yn) .

2.2 The im'son’s Rule&imilarly to the tra#e$oidal rule hen using the &im#son’s rule you must 5rst

'egin 'y di*iding the area using #arallel line e?ual distan!e a#art from one another.Rather than !losing these sha#es to form tra#e$oid e ill !a# them ith ar!s. "t isim#ortant to note that these ar!s are not that of a !ir!le 'ut of a #ara'ola so thismethod is most a##ro#riate for !al!ulating the area under a !ur*e. The sum of allthese areas gi*es you an a##ro6imate area. -eri*ing the e?uation for the &im#son’srule in*ol*es mathemati!s far outside the realm of my one #ersonal !a#a'ility so forthe #ur#ose of this re#ort it ill 'e sim#ly stated as9

A= h3

( y0 +4 y1 +2 y2 +4 y3 …+2 yn−2 +4 yn− 1 + yn) .

2.3 Areas ,y &ntegration The #re*iously t o dis!ussed methods the summation of areas !an 'e used

to 5nd an a##ro6imate area under a !ur*e. Co e*er integration uses a similar#remise to !al!ulate a more e6a!t area. "ntegration is essentially the summation ofan in5nite amount of #arallel lines 'et een t o 'ound. The summation of the areasof all these lines deri*es an e6a!t area. Thus in mathemati!s the integral of a gi*en

fun!tion is the area under its !ur*e."f one !onsiders the idth of one of these #arallel lines to 'e d6 then the

length of the line is !onsidered to 'e the yH!oordinate this line interse!ts ith on the!ur*e. The summation of areas of all these lines 'et een any t o #oints on a gra#h

!an then 'e !al!ulated 'y the e?uation9 A= ∫a

b

ydx or A=∫a

b

f ( x)dx . This

e?uation is suita'le for !al!ulating areas *erti!ally ho e*er hen a !ur*e is su!hthat the area must 'e !al!ulated hori$ontally then the 6H!oordinate 'e!omes thelength of ea!h line and the distan!e y is its idth so the e?uation for 5nding the

area 'e!omes9 A=∫a

b

xdy or A=∫a

b

g ( y)dy .


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3.0 Measurements for Cal!ulating -olume To this #oint e ha*e not dis!ussed three dimensional o'>e!ts. 8ne any sha#e

is gi*en a third dimension its *olume !an 'e !al!ulates. This re#ort ill introdu!ethe most !ommonly =no n three dimensional sha#es other ise =no n as solids.

The most !ommon solid 5gure is the re!tangular solid. This 5gure has si6 sides=no n as fa!es ea!h of these fa!es is re!tangular in sha#e. Another !ommonly=no n solid is the !ylinder. The !ylinder is sim#ly a re!tangle rotated a'out one ofits sides su!h that its 'ase is !ir!ular. /e6t is the !ir!ular !one. A !ir!ular !one isgenerated 'y rotating a right angle triangle a'out one of its legs. &imilar to the !oneis the #yramid. The !one and #yramid share similar ?ualities 'ut a #yramid’s 'ase isre!tangular rather than !ir!ular and its sides are triangular. Einally e ha*e thes#here. A s#here is generated 'y rotating a !ir!le a'out its diameter. The!al!ulations for surfa!e area and *olume of these solids are gi*en 'y the follo ingta'le9

Figure 3" our!e )ashington '. 2 !h.2*


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3.1 -olumes ,y &ntegration

3.1.1 -olume of a Cur/eAs dis!ussed in an earlier se!tion F3.0 Measurements for al!ulating (olumeG

an o'>e!t 'e!omes a solid hen rotated a'out a #oint of a6is. This is !on!e#tually

true hen a fun!tion is de#i!ted gra#hi!ally. The *olume of a !ur*e !an 'e!al!ulated su!h that the !ross se!tional area dx or dy is rotated a'out its

res#e!ti*e a6is !reating a three dimensional solid. The *olume of a fun!tion is

deri*ed from its integral using the formula V = π ∫a

b

y2dx = π ∫

a

b

[f ( x)]2 dx for *olumes

re*ol*ing a'out the 6Ha6is and V = π ∫a

b

x2dy= π ∫

a

b

[f ( x)]2 dy for *olumes re*ol*ing

a'out the yHa6is.

3.1.2 -olume of a hell There are !ertain fun!tions that hen re*ol*ed

a'out their res#e!ti*e a6is they do not !reate a dis! 'utrather a !ylindri!al shell. This !ylindri!al shell is !reated'y the formation of t o !on!entri! ith its gi*en height

either y or x . The *olume of these fun!tions is made

u# of the summation of an in5nite num'er of !ylinders'et een the t o 'ounds stated. &in!e e ha*e noted in earlier se!tions F1.4 ir!lesGthe area of a !ir!le then 'y multi#lying this e?uation 'y a height and ta=ing itsintegral e !an !al!ulate its *olume. &imilar to the e?uation in the #re*ious se!tion

the *olume of a !ylindri!al shell !an 'e !al!ulated using the formula V = 2 π ∫a

b

xydx

here 6 is its radius y is its height and d6 is its thi!=ness.


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%.0 Trigonometry to ol/e for Cantile/ers

Figure % our!e" la! ,oard learning materials

I/ote9 -e#ending on =no n sides and angles of the gi*en triangle the other

res#e!ti*e trigonometri! fun!tions sin θ∨cos θ !an 'e used to sol*e in the same

e?uations to sol*e for un=no n *aria'les.


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.0 Centroids"n mathemati!s #arti!ularly me!hani!s the

!entroid is an o'>e!t’s !enter of mass. @efore

dis!ussing !entroids e must 5rst 'e!ome familiarith the !on!e#t of moments. A moment is the

tenden!y of an o'>e!t to rotate a'out a #oint. &in!ea moment is for!e it !an 'e !al!ulated 'y multi#lying

its mass times a distan!e from or F = md .

"f e are to #re*ent an o'>e!t from rotatinga'out a #oint e must 5nd its !enter of mass or

!entroid hi!h is the a #oint d́ units a ay from the gi*en #oint here all the

masses !ould 'e !on!entrated to 5nd an identi!al total moment. The 'alan!ing ofan e?uation to 5nd o'>e!ts !entroid !an then 'e noted as9

m1 d 1 +m1 d1 +…+m1 d 1 =( m1 +m2 +…+mn) d́

.0 imensionless 4rou'sith regard to mathemati!s and engineering #rin!i#les a dimensionless

grou# is a ?uantity to hi!h no #hysi!al dimension is rele*ant it is often referred to

as a dimension of one. Jntities su!h as π ,e ,∨ are !ommonly used num'ers

hi!h !an 'e regarded as dimensionless. To gi*e some #ers#e!ti*e on the matter?uantities hi!h are no dimensionless are *alues su!h as time or distan!e hi!h!an 'e measured in se!onds or meters. A 5nal note to ma=e ith res#e!t todimensionless grou#s is that they are ty#i!ally ratio or #rodu!ts of other ?uantities.&train for instan!e is the deri*ed 'y di*iding a !hange in length 'y its initial length.8n!e this #ro!ess is a##lied the units ill !an!el ea!h other out and strain is

therefore dimensionless. ! = " l /l# .

.1 Reynold’s 5um,erReynold’s num'er is used in Duid me!hani!s to #redi!t Do #atterns in gi*en

s!enarios. The te!hni!al de5nition of a Reynolds num'er is the ratio of inertial

for!es to *is!ous for!es to !al!ulate Do !onditionsB. The a##li!ation of theReynold’s num'er formula is ty#i!ally used to !ategori$e a Duids Do as laminar ortur'ulent. Laminar Do is Do su!h that the Reynold’s num'er is lo thus the*is!ous for!es are greater than that of the inertial for!es. Tur'ulent Do is then theresult of a high Reynold’s num'er su!h that the inertial for!es are dominant.

Figure our!e"


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ℜ= #ner#al f$rces%#sc$&sf$rces

= '%L ( here * is *elo!ity l is !hara!teristi! length

' is the

density and ( is the dynami! *is!osity.

*.0 7nits and 7nit Con/ersionshen dealing ith any #ro'lem in mathemati!s and engineering it is

essential to 'oth identify and unify all units. @elo are t o sour!es imagesidentifying !ommonly used units as ell as !on*ersions 'et een metri! andim#erial units9


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Figure * our!e" ###.lhu'.edu


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i,liogra'hy

K1 A. . ashington N eometry N in Technical Mathmatic 10th Edition hi!agoPeason 2013 ##. %,H:+2.

Algon8uin College


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Center for Constru!tion 9:!ellen!e

e'artment of Ar!hite!ture; Ci/il and uilding!ien!e

C 2300 A''lied Mathemati!s

Case Study # 1 Area, Volume, and Energy Calculations in Buildings

/ame9 &!ott Rogers

&tudent /um'er9 040%:%%),Eor9 -r. A'dul AlHA$$a i

-ate9 Mar!h 14 201)

Area Volume Calculation in Structures

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