Regular hexagons are inscribed in and circumscribed outside a circle.

If the smaller hexagon has an area of three square units, what is the area of the larger hexagon?

Source: Charles W. Trigg, Mathematical Quickies.

Let the radius of the circle be r.

Consider 1/6 of the hexagon, which is an equilateral triangle.

The altitude of the larger triangle is r.

The side of the smaller triangle is also r, which makes its altitude r/2*Sqrt(3).

The ratio of the larger to the smaller altitude is 1:Sqrt(3)/2,

or 2:sqrt(3).

Squaring this, the ratio of the areas is 4:3.

So the larger hexagon = 4 square units.