Question

the area of a sector of a circle witha radius measuring 30cm 100 pi cm. what is the measure of the central angle that forms the sector?​

Answer 1

The central angle is
`40\°` or
`(2)/(9)\pi\ radians`

Explanation:

step 1

Find the area of the circle

The area of the circle is equal to

`A=\pi r^(2)`

we have

`r=30\ cm`

substitute

`A=\pi (30)^(2)`

`A=900\pi\ cm^(2)`

step 2

Find the central angle in degrees for a sector with area
`100\pi\ cm^(2)`

Let

x----> the measure of the central angle in degrees

Remember that the area of the circle subtends a central angle of 360 degrees

so

using proportion

`(900\pi)/(360)=(100\pi)/(x)\\ \\x=360*100\pi/900\pi \\ \\x=40\°`

step 3

Find the central angle in radians for a sector with area
`100\pi\ cm^(2)`

Let

x----> the measure of the central angle in radians

Remember that the area of the circle subtends a central angle of
`2\pi` radians

so

using proportion

`(900\pi)/(2\pi)=(100\pi)/(x)\\ \\x=2\pi*100\pi/900\pi \\ \\x=(2)/(9)\pi\ radians`