Finding the measure of an angle.

Geometry Level 1

Circle O O has a radius of 5 5 . If the area of the yellow region is 20 π 20\pi , find x x (in degrees).


The answer is 18.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Consider the diagram on the left.

A s e c t o r = A c i r c l e − A y e l l o w r e g i o n = π ( 5 2 ) − 20 π = 25 π − 20 π = 5 π A_{sector}=A_{circle}-A_{yellow~region}=\pi (5^2)-20 \pi=25 \pi -20\pi=5 \pi

So,

β 360 π r 2 = 5 π \dfrac{\beta}{360} \pi r^2=5 \pi

β = 5 ( 360 ) 25 = 72 \beta=\dfrac{5(360)}{25}=72

Finally, x = 90 − 72 = x=90-72= 18 \boxed{18}

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