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 60 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.
+128475 This doesn't seem possible...I believe this should read "The sides of a 45 degree angle at P form an arc on the larger circle that is the same length as an arc on the smaller circle formed by the sides of a 60 degree angle at P.
On the larger circle 45° = pi/4 ... On the smaller circle 60° = pi/3
Arc length on smaller circle = r(pi/ 3) where r is the radius of the smaller circle
Arc length on larger circle = R (pi/4) where R is the radius of the larger circle
And since the arc lengths are equal
r (pi/3) = R (pi/4)
r = R (pi/4) / (pi/3)
r = (3/4)R
Area of smaller circle = pi ( (3/4)R)^2 9
_________________ _____________ = ___
Area of larger circle pi R^2 16
