Unit 27

AREAS OF CIRCLES, SECTORS, SEGMENTS, AND ELLIPSES

2

AREAS OF CIRCLES

The area of a circle is equal to the product of and the square of the radius (A = r2)

The areas of two circles have the same ratio as the squares of the radii or diameters

22

21

22

21

2

1

d

d

r

r

A

A

3

AREAS OF CIRCLES

The areas of two circles are 144 mm2 and 36 mm2. Compare the radius of the larger circle with the radius of the smaller circle:

AnsA

A

r

r

r

r

A

A2

36

144;Since

2

1

2

12

2

21

2

1

– The radius of the larger circle is 2 times larger than the radius of the smaller circle

4

AREAS OF SECTORS

A sector of a circle is a figure formed by two radii and the arc intercepted by the radii

The area of a sector is given as:

2π360

θrA

5

AREAS OF SECTORS

The area of a sector is given as:

Determine the area of a piece of pizza with a central angle of 48° and a radius of 10inches

2π360

θrA

AnsA22 in41.8910π

360

48

6

AREAS OF SEGMENTS

A segment of a circle is a figure formed by an arc and the chord joining the end points of the arc

The area of a segment is found by subtracting the area of a triangle from the area of a sector That triangle is always isosceles

2 radii for sides

7

AREAS OF SEGMENTS Find the area of segment ACB in the

figure below, given that AOB = 85, the radius of the circle is 4 in, and AB is 10 in:

– Area of the sector:A = (85°/360°)(4")2 = 11.868 in2

– Area of the triangle:A = ½ bh = ½ (10")(2") = 10 in2

– Area of the segment:A = 11.868 in2 – 10 in2 = 1.868 in2 Ans

C

BA

O

2"

8

AREAS OF SEGMENTS

You build a nice deck in your rectangular back yard that is circular right outside the sliding doors. The yard is 50 ft long and 100 ft wide. The sides by the patio are 15 and 30 feet. The angle for the patio is 135° and the radius is 31.5 ft. Find the grass area in the yard.

25000

50*100

ftA

A

YardofArea

9

AREAS OF SEGMENTS

You build a nice deck in your rectangular back yard that is circular right outside the sliding doors. The yard is 50 ft long and 100 ft wide. The sides by the patio are 15 and 30 feet. The angle for the patio is 135° and the radius is 31.5 ft. Find the grass area in the yard.

So the yard is 5000 square feet without a deck so what does the deck take out?

2

2

97.1168

5.31360

135

ftA

A

SectorofArea

2

222

4.422

36.15552

1

36.15

5.315.27

ftA

A

ftb

b

heightaneed

TriangleofArea

10

AREAS OF SEGMENTS

You build a nice deck in your rectangular back yard that is circular right outside the sliding doors. The yard is 50 ft long and 100 ft wide. The sides by the patio are 15 and 30 feet. The angle for the patio is 135° and the radius is 31.5 ft. Find the grass area in the yard.

So the area of the sector minus the area of triangle will give us the segment or the deck.

1168.97 – 422.4 = 746.57 square feet 5000 – 746.57 = 4256.43 square feet of grass remaining after the

desk is installed

11

AREAS OF ELLIPSES

An ellipse is a closed oval-shaped curve that is symmetrical to two lines or axes that are perpendicular to each other

The longer axis is called the major axis and the shorter axis is called the minor axis

The area of an ellipse is equal to the product of and one half the major axis and one half the minor axis

12

AREAS OF ELLIPSES

Find the surface area of an elliptical dining table that is 8 feet long (major axis) and 5 feet wide (minor axis):

Area = (8 ft 2)(5 ft 2)

= 31.416 ft2 Ans

13

PRACTICE PROBLEMS

1. Find the area of a circle that has a radius of 7.25 meters.

2. Determine the radius of a circular patio which is to have an area of 14 square yards.

3. The radii of two circles are 6 inches and 2 inches. Compare the area of the larger circle with the area of the smaller circle.

4. Find the area of the sector of a circle with a central angle of 78° and a radius of 4.5 inches.

14

PRACTICE PROBLEMS (Cont)

5. Determine the central angle for a sector of a circle with a 3-meter radius given that the area of the sector is 3.77 square meters.

6. Find the area of a segment of a circle given a central angle of 60° and a radius of 4 inches when the height and base of the triangular section are 2 inches and 3 inches respectively.

7. Determine the area of an ellipse with a major axis of 7.5 cm and a minor axis of 5.5 cm.

8. Determine the major axis of an ellipse if its area is 236.34 square yards and the minor axis is 12.3 yards.

15

9. In a circular tank there is this cross sectional view and measurements. What is the area of the water in the bottom of the tank?

PRACTICE PROBLEMS (Cont)

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PROBLEM ANSWER KEY1. 165.13 m2

2. 2.111 yards3. The area of the larger circle is 9 times

larger than the area of the smaller circle

4. 13.784 in2

5. 48°6. 5.378 in2

7. 32.4 cm2 8. 24.465 yards9. 17.39 yards2