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Find the central angle of a sector of a circle of the area of the sector and the area of the circle are in the proportion of 3:5

User Sbonkosky
by
8.8k points

1 Answer

4 votes

Answer:


\theta = 216

Explanation:

Given

Area of Sector : Area of Circle = 3 : 5

Required

Determine the central angle

The question implies that


(Area_(sector))/(Area_(circle)) = (3)/(5)

Multiply both sides by 5


5 * (Area_(sector))/(Area_(circle)) = (3)/(5) * 5


5 * (Area_(sector))/(Area_(circle)) = 3

Multiply both sides by Area{circle}


5 * (Area_(sector))/(Area_(circle)) * Area_(circle) = 3 * Area_(circle)


5 * {Area_(sector) = 3 * Area_(circle)

Substitute the areas of sector and circle with their respective formulas;


Area_(sector) =(\theta)/(360) * \pi r^2


Area_(circle) = \pi r^2

So, we have


5 * (\theta)/(360) * \pi r^2 = 3 * \pi r^2

Divide both sides by
\pi r^2


5 * (\theta)/(360) * ( \pi r^2)/(\pi r^2) = 3 * (\pi r^2)/(\pi r^2)


5 * (\theta)/(360) = 3

Multiply both sides by 360


360 * 5 * (\theta)/(360) = 3 * 360


5 * \theta = 3 * 360

Divide both sides by 5


(5 * \theta)/(5) = (3 * 360)/(5)


\theta = (3 * 360)/(5)


\theta = (1080)/(5)


\theta = 216

Hence, the central angle is 216 degrees

User Quinestor
by
8.8k points

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