Inscribed triangle and circle

Geometry Level 3

In the figure on the right, a square and an equilateral triangle are inscribed in a circle. The ratio of the square's area and the triangle's area can be written as $\frac{a}{b\sqrt{b}}$ , where $a$ and $b$ are coprime integers. Find $a+b$

The answer is 11.

1 solution

A Former Brilliant Member
Jun 17, 2018

Let $r$ be the radius of the circle. The area of the inscribed equilateral triangle is $\dfrac{3\sqrt{3}}{4}r^2$ . The area of the inscribed square is $2r^2$ . The ratio of the area of the square to the area of the equilateral triangle is

$\dfrac{2r^2}{\dfrac{3\sqrt{3}}{4}r^2}=2 \times \dfrac{4}{3\sqrt{3}}=\dfrac{8}{3\sqrt{3}}$

The desired answer is $a+b=8+3=\boxed{11}$

Inscribed triangle and circle

The answer is 11.

1 solution

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