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.
Rotate the inner hexagon so its verticies meet the tangent midpoints of the outer hexagon. A set of six 30-30-120 triangles is formed, representing the difference in area.
Then divide the inner hexagon into 6 equilateral triangles. Divide each of those triangles into three 30-30-120 triangles. All 18 of these new triangles are congruent to the 6 triangles formed earlier.
Then the ratio of area is 24:18 = 4:3. The larger hexagon has an area of four square units.