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'Funky' Circle Areas

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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!

Apr 18, 2018

#1
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Let the radius of the small circle  be r

Let the radius of the larger circle be R

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   Apr 18, 2018
#2
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Thanks CPhill!

AnonymousConfusedGuy  Apr 18, 2018