Ratio

Geometry Level 2

A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon.

\dfrac{4}{3}

\dfrac{7}{4}

\dfrac{5}{3}

\dfrac{3}{2}

1 solution

Marta Reece
Jun 15, 2017

If the side of the outside hexagon is $1$ , then the radius of the inscribed circle $r^2=1-\left(\dfrac12\right)^2$ and $r=\dfrac{\sqrt3}{2}$

The side of the smaller hexagon is equal to the radius of the circle it is inscribed in, so it is also $\dfrac{\sqrt3}{2}$

The ration of area is the square of the ratio of sides: $\dfrac{1}{\left(\dfrac{\sqrt3}{2}\right)^2}=\boxed{4:3}$

Ratio

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