Chapter 11- Areas of Plane Figures By Lilli Leight, Zoey Kambour, and Claudio Miro.
-
Upload
herbert-osborne -
Category
Documents
-
view
221 -
download
0
Transcript of Chapter 11- Areas of Plane Figures By Lilli Leight, Zoey Kambour, and Claudio Miro.
Chapter 11-Areas of Plane FiguresBy Lilli Leight, Zoey Kambour, and Claudio Miro
11.1-Areas of Rectangles Postulate 17-The area of a square is the
square of the length of a side. A=S2
3
3
Length: 1 unit Area: 1 square unitBy counting, Area=9 square unitsBy using the formula, Area=32=9
11.1-Areas of Rectangles Postulate 18 (Area Congruence
Postulate)-If two figures are congruent, then they have the same area.
Postulate 19 (Area Addition Postulate)-The area of a region is the sum of the areas of its non-overlapping parts.
11.1-Areas of Rectangles Theorem 11.1-The area of a rectangle
equals the product of its base and height. A=bh
A H2
B2 A
b h
h
b
h
b
b h
Given: A rectangle with base b and height hProve: A=bhProof:Draw-the given rectangle with area A, a congruent rectangle with area A, a square with area b2, a square with area h2
Area of big square= 2A+b2+h2
Area of big square= (b+h)2= b2+2bh+h2
2A+b2+h2 = b2+2bh+h622A = 2bh A = bh
11.1 Practice Problems What is the area of a rectangle with a
base of 5 and a height of 7? What is the area of a square with a base
of 4 and a height of 4?
11.2-Areas of Parallelograms, Triangles, and Rhombuses Theorem 11.2-The area of a
parallelogram equals the product of a base and the height to that base A=bh
S R
QP
h
V
I II III
b
h
TGiven: PQRSProve: A=bhKey steps of proof:1.) Draw altitudes PV and QT, forming two rt, triangles2.)Area I=Area III 3.) Area of PQRS= Area II + Area I
= Area II + Area III = Area of rect. PQTV = bh
11.2 Areas of Parallelograms, Triangles, and Rhombuses Theorem 11.3-The area of a triangle
equals half the product of a base and the height to that base A=1/2bh
h
W
ZY
b
Given: XYZProve: A=1/2bhKey steps of proof:1.) Draw XW parallel to YZ and ZW parallel to YX forming WXYZ2.) XYZ congruent to ZWX (SAS or SSS)3.) Area of XYZ = ½ x Area of WXYZ
=1/2bh
X
11.2-Areas of Parallelograms, Triangles, and Rhombuses 11.4-The area of a rhombus equals half
the product of its diagonals. A=1/2d1d2
Given: Rhombus ABCD with diagonals d1
and d2
Prove: A=1/2d1d2
Key steps of proof:1.) ADC congruent ABC (SSS)2.)Since DB is perpendicular to AC, the area of ADC = ½ bh = ½ x d1 x 1/2d2=1/4d1d2
3.)Area of rhombus ABCD=2 x 1/4d1d2= 1/2d1d2
1/2d2
1/2d2
d1
DC
BA
11. 2 Examples1.) Find the area of a parallelogram with sides 8 and 15, and the acute angle equal to 35 degrees.
Sin(35)=X/8X=4.588 x 15 = 68.83 8
35
152.) The area of a triangle is 410 with a base of 41. Find its height. A=410 A=1/2bh 410=1/2(41)(h) 410=(20 x 5)(h) h=20
11.2 Practice Problems Find the area of a parallelogram with sides 6
cm and 8 cm, and a 135 degree angle. A rhombus has a perimeter of 60 and one
diagonal of 24. Find its area. Find the area of:
1.) An equilateral triangle with a perimeter of 24 2.) An isosceles triangle with sides 13, 13, and
10. 3.) 30-60-90 triangle with a hypotenuse of 12
inches.
11.3 Area of a Trapezoid Theorem 11.5-The area of a trapezoid
equals half the product of the height and the sum of the bases. (A=1/2h(b1+b2)) Also A=h(median)
A
D C
B
II
I
b1
b2
hh
Key steps of proof:1.) Draw a diagonal BD of trap. ABCD,forming two triangular regions, I and II, each with a height of h.2.) Area of a trapezoid = Area I+ Area II
= 1/2b1h+1/2b2h = 1/2h(b1+b2)
11.3 Example Find the area of a trapezoid with a
height of 7 and a median of 15. 15 x 7=105
15
7
11.3 Practice Problem A trapezoid has an area of 75 cm2 and a
height of 5 cm. How long is the median?
5 cm
11.4 Regular Polygon A regular Polygon is any convex shape
whose sides are all the same length and angles are all the same measure.
11.4 Regular Polygons
Center of a regular polygon= Center of circumscribed circle.
Radius of a regular polygon= distance from a center to a vertex.
Central Angle of a regular polygon= an angle formed by 2 radii drawn to consecutive vertices.
Apothem of a regular polygon= Perpendicular distance from the center to one of the sides of the polygon.
• Regular Polygons can be inscribed inside of a circle. Using this relationship, some definitions were derived.
11.4 Practice Problems Find the perimeter and the area of each
figure.1. A regular octagon with sides 4 and
apothem a.2. A regular pentagon side s and apothem
of 3. 3. A regular decagon with side s and
apothem a.5
9
11.4 Theorem 11-6 The area of a regular polygon is one half
of the product of the apothem and the perimeter. A=1/2*P*a
P=Perimeter a= Apothem
Ex: P=10*6=60 a=8.66 A=(1/2)*60*8.66=259.8 sq. units
10
8.66
11.5 Circumference Circumference: Perimeter of a circle.
It is found by the product of twice the radius (diameter) and pi.
C= 2πr or C= πd C= circumference r= radius d=
diameter Ex: Find the circumference of a circle with a
radius of 12. C=? r= 12 C= (2)π(12)= 24π units
11.5 Area Area= As the area of the inscribed
regular polygons get closer and closer to a limiting number defined to be the area of a circle. It is found by the product of the radius
square and pi. A= πr2 r = radius
Ex: Find the area of a circle with a radius of 27 A=? r= 27 A= (272) π = 729π sq. units
11.5 Practice Problems1. A circle has an area of 18 in. Find the
circumference of the rim.2. Find the area of a circle with a radius
of 73. Find the radius and area of a circle with
a circumference of 20π.4. Find the radius and circumference of a
circle with an area of 25π
11.6 Arc Lengths and Areas of Sectors Sector- region bounded by two radii and
an arc of the circle.
11.6 Finding the length of a sector• Length of sector: x/360(2πr)
x = number of degrees in sector r = radius Ex. Find the length of the sector
120/360(2x9π)1/3(18 π) = 6 π
1209
11.6 Finding the area of a sector Area of sector: x/360 = πr2
x = degree of sector r = radiusEx. Find the area of the sector
45/360(42 π)5/72(16 π)= 10 π/9
45
4
11.6 Finding the length of the radii If you are only given the area or length of a
sector and the sector degree measurement, you can still find the length of the radius.
Practice problems1. Find the length of the radius with length of
the sector is 2π and the degree measurement is 90.
2. Find the length of the radius with length of sector being 4π and area of sector 120.
11.6 Practice Problems Find the area and the length of each
sector
1
1
11.7 Ratios of Areas Theorem 11-7: If the scale factor of two
similar figures a:b, then The ratio of the perimeters is a:b The ratio of the areas is a2:b2 Ex. Find the ratio of the perimeters and the
areas of the two similar figures. The scale
factor is 8:12 or 2:3.Therefore, the
ratio of the perimeter is 2:3. The ratio of the areas is 23:32 or
4:9.
8 12
11.7 Practice Problems If the scale factor is 1:4, what is the ratio
of the perimeter and the ratio of the areas?
If the ratio of the areas I 25:1, what is the ratio of the perimeter and the scale factor?
A quadrilateral with sides 8cm, 9cm, 6cm and 5cm has area 45 cm2. Find the area of a similar quadrilateral who's longest side is 15 cm.