We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website.
Please click on "Accept cookies" if you agree to the setting of cookies. Cookies that do not require consent remain unaffected by this, see
cookie policy and privacy policy.
DECLINE COOKIES

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