One outside, one inside

Geometry Level 3

There is an equilateral triangle of side length $r$ . 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.

1 solution

Mehul Arora
Feb 25, 2016

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

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

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

Thus, $R=2r$

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

One outside, one inside

The answer is 0.25.

1 solution

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