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Area and Circumference4.4

4.4 OBJECTIVES

1. Use p to find the circumference of a circle2. Use p to find the area of a circle3. Find the area of a parallelogram4. Find the area of a triangle5. Convert square units

In Section 4.2, we again looked at the perimeter of a straight-edged figure. The distancearound the outside of a circle is closely related to this concept of perimeter. We call theperimeter of a circle the circumference.

341

Example 1

Finding the Circumference of a Circle

A circle has a diameter of 4.5 ft, as shown in Figure 2. Find its circumference, using 3.14for p. If your calculator has a key, use that key instead of a decimal approximationfor p.

p

The circumference of a circle is the distance around that circle.

Definitions: Circumference of a Circle

Let’s begin by defining some terms. In the circle of Figure 1, d represents the diameter.This is the distance across the circle through its center (labeled with the letter O, for ori-gin). The radius r is the distance from the center to a point on the circle. The diameter isalways twice the radius.

It was discovered long ago that the ratio of the circumference of a circle to its diameteralways stays the same. The ratio has a special name. It is named by the Greek letter p (pi).

Pi is approximately , or 3.14 rounded to two decimal places. We can write the following

formula.

22

7

O

d

Radius

Diameter

Circumference

Figure 1

C � pd (1)

Rules and Properties: Formula for the Circumference of a Circle

NOTE The formula comes fromthe ratio

� pCd

4.5 ft

Figure 2

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C � 2pr (2)

Rules and Properties: Formula for the Circumference of a Circle

Example 2

Finding the Circumference of a Circle

A circle has a radius of 8 in., as shown in Figure 3. Find its circumference using 3.14 for p.

From Formula (2),

C � 2pr

� 2 � 3.14 � 8 in.

� 50.2 in. (rounded to one decimal place)

8 in.

Figure 3

NOTE Because d � 2r (thediameter is twice the radius)and C � pd, we have C � p(2r),or C � 2pr.

C H E C K Y O U R S E L F 2

Find the circumference of a circle with a radius of 2.5 in.

NOTE Because 3.14 is anapproximation for pi, we canonly say that the circumferenceis approximately 14.1 ft. Thesymbol � means approximately.

NOTE If you want toapproximate p, you needn’tworry about running out ofdecimal places. The value for pihas been calculated to over100,000,000 decimal places on acomputer (the printout wassome 20,000 pages long).

C H E C K Y O U R S E L F 1

A circle has a diameter of inches (in.). Find its circumference.312

Note: In finding the circumference of a circle, you can use whichever approximation for piyou choose. If you are using a calculator and want more accuracy, use the key.

There is another useful formula for the circumference of a circle.p

By Formula (1),

C � pd

� 3.14 � 4.5 ft

� 14.1 ft (rounded to one decimal place)

AREA AND CIRCUMFERENCE SECTION 4.4 343©

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Example 3

Finding Perimeter

We wish to build a wrought-iron frame gate according to the diagram in Figure 4. Howmany feet (ft) of material will be needed?

The problem can be broken into two parts. The upper part of the frame is a semicircle (halfa circle). The remaining part of the frame is just three sides of a rectangle.

Circumference (upper part)

Perimeter (lower part) � 4 � 5 � 4 � 13 ft

Adding, we have

7.9 � 13 � 20.9 ft

We will need approximately 20.9 ft of material.

�1

2� 3.14 � 5 ft � 7.9 ft

5 ft

4 ft

Figure 4

NOTE Using a calculator with akey,

1 2 5�p��

p

Sometimes we will want to combine the ideas of perimeter and circumference to solvea problem.

C H E C K Y O U R S E L F 3

Find the perimeter of the following figure.

6 yd

8 yd

The number pi (p), which we used to find circumference, is also used in finding the areaof a circle. If r is the radius of a circle, we have the following formula.

NOTE The distance around the

semicircle is 12

pd.

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Example 4

Find the Area of a Circle

A circle has a radius of 7 inches (in.) (see Figure 5). What is its area?

Use Formula (3), using 3.14 for p and r � 7 in.

A � 3.14 � (7 in.)2

� 153.86 in.2

7 in.

Figure 5

Again the area is an approximation because we use 3.14, an approximationfor p.

C H E C K Y O U R S E L F 4

Find the area of a circle whose diameter is 4.8 centimeters (cm). Remember that theformula refers to the radius. Use 3.14 for p, and round your result to the nearesttenth of a square centimeter.

Two other figures that are frequently encountered are parallalograms and triangles.

In Figure 6 ABCD is called a parallelogram. Its opposite sides are parallel and equal.Let’s draw a line from D that forms a right angle with side BC. This cuts off one corner ofthe parallelogram. Now imagine that we move that corner over to the left side of the figure,as shown. This gives us a rectangle instead of a parallelogram. Because we haven’t changedthe area of the figure by moving the corner, the parallelogram has the same area as therectangle, the product of the base and the height.

A

B C

Db

h

Figure 6

A � pr2 (3)

Rules and Properties: Formula for the Area of a Circle

NOTE This is read, “Areaequals pi r squared.” You canmultiply the radius by itself andthen by pi.

AREA AND CIRCUMFERENCE SECTION 4.4 345

Another common geometric figure is the triangle. It has three sides. An example istriangle ABC in Figure 8.

b is the base of the triangle.h is the height, or the altitude, of the triangle.

Once we have a formula for the area of a parallelogram, it is not hard to find the area of atriangle. If we draw the dotted lines from B to D and from C to D parallel to the sides ofthe triangle, we form a parallelogram. The area of the triangle is then one-half the areaof the parallelogram [which is b � h by Formula (4)].

A

B D

C

h

b

Figure 8

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A � b � h (4)

Rules and Properties: Formula for the Area of a Parallelogram

Example 5

Finding the Area of a Parallelogram

A parallelogram has the dimensions shown in Figure 7. What is its area?

Use Formula (4), with b � 3.2 in. and h � 1.8 in.

A � b � h

� 3.2 in. � 1.8 in. � 5.76 in.2

1.8 in.

3.2 in.

Figure 7

C H E C K Y O U R S E L F 5

If the base of a parallelogram is in. and its height is in., what is its area?112

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C H E C K Y O U R S E L F 6

A triangle has a base of 10 feet (ft) and an altitude of 6 ft. Find its area.

Sometimes we will want to convert from one square unit to another. For instance, lookat 1 yd2 in Figure 10.

The table below gives some useful relationships.

1 yd = 3 ft 1 yd = 9 ft2 2

1 yd = 3 ft

Figure 10

Square Units and Equivalents

1 square foot (ft2) � 144 square inches (in.2)1 square yard (yd2) � 9 ft2

1 acre � 4840 yd2 � 43,560 ft2NOTE Originally the acre wasthe area that could be plowedby a team of oxen in a day!

Example 6

Finding the Area of a Triangle

A triangle has an altitude of 2.3 in., and its base is 3.4 in. (see Figure 9). What is its area?

Use Formula (5), with b � 3.4 in. and h � 2.3 in.

�1

2� 3.4 in. � 2.3 in. � 3.91 in.2

A �1

2 � b � h

2.3 in.

3.4 in.

Figure 9

A � (5)12

� b � h

Rules and Properties: Formula for the Area of a Triangle

AREA AND CIRCUMFERENCE SECTION 4.4 347

Example 8 illustrates the use of a common unit of area, the acre.

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Example 7

Converting Between Feet and Yards in Finding Area

A room has the dimensions 12 ft by 15 ft. How many square yards of linoleum will beneeded to cover the floor?

A � 12 ft � 15 ft � 180 ft2

20

� 180 ft2 �

� 20 yd2

1

9 yd2

ft2

121 �

NOTE We first find the area insquare feet, then convert tosquare yards.

C H E C K Y O U R S E L F 7

A hallway is 27 ft long and 4 ft wide. How many square yards of carpeting will beneeded to carpet the hallway?

Example 8

Converting Between Yards and Acres in Finding Area

A rectangular field is 220 yd long and 110 yd wide. Find its area in acres.

A � 220 yd � 110 yd � 24,200 yd2

� 24,200 yd2

� 5 acres

1

4840 acre

yd2

C H E C K Y O U R S E L F 8

A proposed site for an elementary school is 220 yd long and 198 yd wide. Find itsarea in acres.

5

1

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C H E C K Y O U R S E L F A N S W E R S

1. C � 11 in. 2. C � 15.7 in. 3. P � 31.4 yd 4. �18.1 cm2

5. 6. 7. 12 yd2 8. 9 acres

= 30 ft2

� 51

4 in.2

�7

2 in. �

3

2 in.

A �1

2� 10 ft � 6 ftA � 3

1

2 in. � 1

1

2 in.

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Exercises

Find the circumference of each figure. Use 3.14 for p, and round your answer to onedecimal place.

1. 2.

3. 4.

In exercises 5 and 6, use for p, and find the circumference of each figure.

5. 6.

Find the perimeter of each figure. (The curves are semicircles.) Round answers to onedecimal place.

7. 8.

9. 10.

10 in.7 ft

4 ft

1 in. 3 in.

7 ft

9 ft

213 ft2

117 in.

22

7

3.75 ft8.5 in.

5 ft9 ft

4.4

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

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Find the area of each figure. Use 3.14 for p, and round your answer to one decimal place.

11. 12.

13. 14.

In exercises 15 and 16, use for p, and find the area of each figure.

15. 16.

Find the area of each figure.

17. 18.

19. 20.

7 in.

5 in.3 yd

4 yd

8 in.

4 in.

4 ft

7 ft

211 in.3 yd2

1

22

7

8 ft

7 yd

12 ft7 in.

ANSWERS

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

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21. 22.

23. 24.

25. 26.

Solve the following applications.

27. Jogging. A path runs around a circular lake with a diameter of 1000 yards (yd).Robert jogs around the lake three times for his morning run. How far has he run?

28. Binding. A circular rug is 6 feet (ft) in diameter. Binding for the edge costs $1.50per yard. What will it cost to bind around the rug?

29. Lawn care. A circular piece of lawn has a radius of 28 ft. You have a bag of fertilizerthat will cover 2500 ft2 of lawn. Do you have enough?

6 yd

7 yd 2 yd

4 ft

2 ft

5 ft

13 yd

13 yd

12 in.

9 in.

6 ft

11 ft

5 ft

8 ft

ANSWERS

21.

22.

23.

24.

25.

26.

27.

28.

29.

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30. Cost. A circular coffee table has a diameter of 5 ft. What will it cost to have the toprefinished if the company charges $3 per square foot for the refinishing?

31. Cost. A circular terrace has a radius of 6 ft. If it costs $1.50 per square foot to pavethe terrace with brick, what will the total cost be?

32. Area. A house addition is in the shape of a semicircle (a half circle) with a radius of9 ft. What is its area?

33. Amount of material. A Tetra-Kite uses 12 triangular pieces of plastic for its surface.Each triangle has a base of 12 inches (in.) and a height of 12 in. How much materialis needed for the kite?

34. Acreage. You buy a square lot that is 110 yd on each side. What is its size in acres?

35. Area. You are making rectangular posters 12 by 15 in. How many square feet ofmaterial will you need for four posters?

36. Cost. Andy is carpeting a recreation room 18 feet (ft) long and 12 ft wide. If thecarpeting costs $15/yd2, what will be the total cost of the carpet?

37. Acreage. A shopping center is rectangular, with dimensions of 550 by 440 yd. Whatis its size in acres?

38. Cost. An A-frame cabin has a triangular front with a base of 30 ft and a height of20 ft. If the front is to be glass that costs $3 per square foot, what will the glass cost?

ANSWERS

30.

31.

32.

33.

34.

35.

36.

37.

38.

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Find the area of the shaded part in each figure. Round your answers to one decimal place.

39. 40.

41. 42.

43. Papa Doc’s delivers pizza. The 8-inch (in.)-diameter pizza is $8.99, and the price ofa 16-in.-diameter pizza is $17.98. Write a plan to determine which is the betterbuy.

44. The distance from Philadelphia to Sea Isle City is 100 miles (mi). A car was driventhis distance using tires with a radius of 14 in. How many revolutions of each tireoccurred on the trip?

45. Find the area and the circumference (or perimeter) of each of the following: (a) a penny (b) a nickel (c) a dime (d) a quarter (e) a half-dollar (f) a silverdollar (g) a Susan B. Anthony dollar (h) a dollar bill (i) one face of the pyramidon the back of a $1 bill.

46. An indoor track layout is shown below.

How much would it cost to lay down hardwood floor if the hardwood floor costs$10.50 per square meter?

20 m

7 m

10 in.

10 in.

20 ft

20 ft

5 ft

6 ft

Semicircle

2 ft

3 ft

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sANSWERS

39.

40.

41.

42.

43.

44.

45.

46.

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47. What is the effect on the area of a triangle if the base is doubled and the altitude iscut in half? Create some examples to demonstrate your ideas.

48. How would you determine the cross-sectional area of a Douglas fir tree (at, say, 3 ftabove the ground), without cutting it down? Use your method to solve the followingproblem:

If the circumference of a Douglas fir is 6 ft 3 in., measured at a height of 3 ft abovethe ground, compute the cross-sectional area of the tree at that height.

49. What happens to the circumference of a circle if you double the radius? If you doublethe diameter? If you triple the radius? Create some examples to demonstrate youranswers.

50. What happens to the area of a circle if you double the radius? If you double thediameter? If you triple the radius? Create some examples to demonstrate youranswers.

Answers1. 56.5 ft 3. 26.7 in. 5. 55 in. 7. 37.1 ft 9. 34.6 ft

11. 153.9 in.2 13. 38.5 yd2 15. 17. 28 ft2 19. 12 yd2

21. 20 ft2 23. 54 in.2 25. 24 ft2 27. 9420 yd29. Yes; area � 2461.8 ft2 31. �$169.56 33. 864 in.2 35. 5 ft22

37. 50 acres 39. 50.2 ft2 41. 86 ft2 43. 45.47. 49. Doubled; doubled; tripled

95

8 yd2

ANSWERS

47.

48.

49.

50.

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