Two concentric circles are centered at point P. The sides of a 45 degree angle at P form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 36 degree angle at P. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your answer as a common fraction.

Thanks!

AnonymousConfusedGuy
Apr 18, 2018

#1**+2 **

Let the radius of the small circle be r

Let the radius of the larger circle be R

45° = pi/4 in rads 36° = pi/5 in rads

And the arc length, S, of the smalller circle can be expressed as

S = r(pi/4)

But this is equal to the arc length of the larger circle that can be expressed as

S = R(pi/5)

So....this implies that

r(pi/4) = R(pi/5

r/4 = R/5 solve for R

R = (5/4)r

So....the area of the smaller circle is

pi * r^2 (1)

And the area of the larger circle is

pi * R^2 = pi * [ (5/4)r ]^2 = pi (25/16)r^2 (2)

So....the ratio of the area of the smaller circle to the larger circle is just (1) / (2) =

[ pi* r^2 ] / [pi * (25/16) r^2 ] =

16 / 25

CPhill
Apr 18, 2018