Question

an arc of $55$ degrees on circle $a$ has the same length as an arc of $40$ degrees on circle $b$. what is the ratio of the area of circle $a$ to the area of circle $b$? express your answer as a common fraction.

Answer 1

64/121

Explanation:

You want the ratio of areas of circle A to circle B if an arc of 55° on circle A has the same length as one of 40° on circle B.

Arc length

Let ra and rb represent the radii of circle A and circle B, respectively. Then the arc lengths are ...

ra·(55°)(π/180°) = rb·(40°)(π/180°) . . . . proportional to radius and angle

This tells us ...

ra/rb = 40/55 = 8/11

Area

The ratio of areas of the two circles is the square of the ratio of their radii.

AreaA/AreaB = (8/11)² = 64/121

The ratio of areas of circle A to circle B is 64/121.

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