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,
Squaring this, the ratio of the areas is 4:3.
So the larger hexagon = 4 square units.