Problem 1 Finding the Area of a Circle

Sports What is the area of the circular region on the wrestling mat?

Since the diameter of the region is 32 ft, the radius is 32 2 , 32 over 2 , comma  or 16 ft.

A = π r 2 Use the area formula .   = π ( 16 ) 2 Substitute  16  for  r .   = 256 π Simplify .   ≈ 804   .2477193 Use a calculator . table with 4 rows and 3 columns , row1 column 1 , eh , column 2 equals pi , r squared , column 3 cap usetheareaformula . . , row2 column 1 , , column 2 equals pi . open 16 close squared , column 3 cap substitute . 16 , for , r . , row3 column 1 , , column 2 equals 256 pi , column 3 cap simplify , . , row4 column 1 , , column 2 almost equal to 804 , .2477193 , column 3 cap useacalculator . . , end table

The area of the wrestling region is about 804   ft 2 . 804 , ft squared , .

✓ Got It?

1. 1. What is the area of a circular wrestling region with a 42-ft diameter?
   2. Reasoning If the radius of a circle is halved, how does its area change? Explain.

Take note Theorem 10-12: Area of a Sector of a Circle

The area of a sector of a circle is the product of the ratio measure of the arc 360 fraction measure of the arc , over 360 end fraction  and the area of the circle.

Area of sector A O B = m A B ⌢ 360 ⋅ π r 2 eh o b equals . fraction m , modified eh b with frown above , over 360 end fraction . dot pi , r squared

Problem 2 Finding the Area of a Sector of a Circle

What is the area of sector GPH? Leave your answer in terms of π . pi .

area of sector  G P H = m G H ⌢ 360 ⋅ π r 2     = 72 360 ⋅ π ( 15 ) 2 Substitute  72   for  m G H ⌢  and  15   for  r .   = 45 π Simplify. table with 3 rows and 3 columns , row1 column 1 , area of sector g p h , column 2 equals . fraction m , modified g h with frown above , over 360 end fraction . dot pi , r squared , column 3 , row2 column 1 , , column 2 equals , 72 over 360 , dot pi . open 15 close squared , column 3 cap substitute . 72 , for , m , modified g h with frown above . and , 15 , for , r . , row3 column 1 , , column 2 equals 45 pi , column 3 cap simplify. , end table

The area of sector GPH is 45 π   cm 2 . 45 pi . , cm squared , .

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2. A circle has a radius of 4 in. What is the area of a sector bounded by a 45 ° 45 degrees  minor arc? Leave your answer in terms of π . pi .