GEOMETRY: Ch. 1.7

1.7: Find Perimeter, Circumference, and

Area

Key Concepts

What is the perimeter?

What is the circumference? 

What is the area?

The perimeter (P) is the distance around a figure.

The circumference (C) is the distance around a circle.

The area (A) is the amount of surface covered by a figure. Perimeter and circumference are measured in units of length such as meters or feet. Area is measured in square units such as square meters.

Formulas for Perimeter P, Area A, and Circumference C

Square s

Side length s

P = 4s

A = s2

Rectangle l

Length l and width w

P= 2l + 2w

A = lw w

Triangle

Side lengths a, b, and c, base b, and height h

P = a + b + c

A = ½ bh

Circle

Diameter d and radius r

C = 2 r

A = r2

Ex. 1. You are trying to find out if you can fit into a ring flotation device. Your waist is 24 inches, and the inner diameter is 12 inches wide.

What do you need to find?

You need to find the circumference.The radius is ½ (12) = 6 = rC = 2πr ≈ 2π(6) ≈ 37.68The circumference is about 37.7 inches. So, the

ring flotation device will fit around your waist.

Ex.2: Find the area and

perimeter of the figure.

The length is 7 inches,

the width is 4.4 inches.

Ex.2: Find the area and

perimeter of the figure.

The length is 7 inches,

the width is 4.4 inches.

A= l w = 7(4.4) = 30.8

The area of the rectangle is 30.8 in2.

The perimeter is

P= 2l + 2w = 2(7) + 2(4.4) = 22.8

The perimeter of the rectangle is 22.8 inches.

Ex. 3Find the area and circumference of the

circle below if the radius is 5 cm.

Ex. 3Find the area and circumference of the

circle below if the radius is 5 cm.

C = 2 r = 2 (5) ≈ 31.4A = r 2 ≈ (5)2 ≈ 79The circumference is about 31.4 cm and

the area is about 79cm2

Ex. 4

Skating Rink—An ice-resurfacing machine is used to smooth the surface of the ice at a skating rink. The machine can resurface about 270 square yards of ice in one minute.

About how many minutes does it take the machine to resurface a rectangular skating rink that is 240 feet long and 100 feet wide?

The machine can resurface about 270 square yards per minute. So, the amount of time it takes to resurface the skating rink depends on its area.

Ex. 4 (cont.)Step 1. Find the area of the rectangular skating rink.

A= l w = 240(100) = 24,000 ft2

The resurfacing rate is in square yards per minute. Rewrite the area of the rink in square yards. There are 3 feet in 1 yard. So, 3 squared feet is equal to nine square feet in 1 square yard.

24,000 ∙ 1/9 = 2,666 square yards

Ex. 4(cont.) Write a verbal model to represent the situation. Then write and solve an equation based on the verbal model.Let t represent the total time (in minutes) needed to resurface the skating rink.

Area of rink = Resurfacing Rate х Total Time(sq. yards) (sq. yards per min) (min)2,666 = 270t9.87 ≈ tIt takes about 9.87 minutes to resurface the rink.

Ex. 5 (#32, p. 54)

The area of a triangle is 27 square feet. The height is three times the length of its base. Find the height and base of the triangle.

Step 1—Write down known variables.

Ex. 5 (#32, p. 54)The area of a triangle is 27 square feet. The

height is three times the length of its base. Find the height and base of the triangle.

Step 1—Write down known variables.A = 27 ft2, h = 3bStep 2—Use the formula for the area of a

triangleA = ½ bh = 27½ b (3b) = 27 Step 3—Substitute 3b for hb2 = 18 Step 4—Solve for b and h

3(3 2) 9 2 feeth 3 2 feetb

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