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

Answer:

The central angle is $40\°$ or $\frac{2}{9}\pi\ radians$

Step-by-step 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

$\frac{900\pi}{360}=\frac{100\pi}{x}\\ \\x=360*100\pi/900\pi \\ \\x=40\°$

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

$\frac{900\pi}{2\pi}=\frac{100\pi}{x}\\ \\x=2\pi*100\pi/900\pi \\ \\x=\frac{2}{9}\pi\ radians$