+1452 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!
+129933 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
