Question

if a central angle of $90$ degrees defines an arc on circle $r$ that has the same length as the arc on circle $w$ defined by a $60$-degree central angle, what is the ratio of the area of circle $r$ to the area of circle $w$? express your answer as a common fraction.

Answer 1

Let R1 be the radius of the first circle

And let R2 be the measure of the second circle

So

Arc length = radius * theta ( in rads )

So....since the arc lengths are equal....

R1 * pi/2 = R2 * pi/3

R1 / 2 = R2 / 3

R1 = (2/3)R2

So...the area of the first circle is

[ pi * (R1) ^2 ] = pi *[ (2/3) R2 ] ^2 = pi (4/9) (R2)^2

And the area of the second circle is

pi [ R2]^2

So the ratio of the area of the first circle to the second is

[ pi * (4/9) (R2)^2 ] 4

_______________ = ____

[ pi * (R2)^2 ] 9

= 4/9

Explanation: