One outside, one inside

Geometry Level 3

There is an equilateral triangle of side length r r . It is inscribed in a circle, and a circle is inscribed inside the triangle. The ratio of the area of the incircle to the area of the circumcircle is?


The answer is 0.25.

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1 solution

Mehul Arora
Feb 25, 2016

We know that Distance between incentre and cicumcentre = d = R ( R 2 r ) = d= \sqrt {R(R-2r)} where R and r are the Circumradius and Inradius respectively. (Thanks , Euler!)

R ( R 2 r ) = 0 \sqrt {R(R-2r)} =0 as the incenter and cicumcenter coincide for an equilateral triangle.

Now since R 0 R \neq 0 , ( R 2 r ) (R-2r) has to be 0, for the distance to be 0.

Thus, R = 2 r R=2r

Ratio of areas= π r 2 π ( 2 r ) 2 = 1 4 = 0.25 \dfrac {\pi r^2}{\pi {(2r)}^2} = \dfrac {1}{4} = 0.25

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