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.