Notes
striped dodecagon solution

Solution to the Striped Dodecagon Puzzle

Striped Dodecagon

This dodecagon is regular. What’s the total shaded area?

Solution by Triangle Area

Striped dodecagon labelled

With the points labelled as above, consider the upper shaded region. This can be decomposed into two triangles, EABE A B and EBDE B D. Triangle EABE A B is congruent to triangle EFBE F B, so the shaded region has the same area as the quadrilateral DEFBD E F B. This region can be decomposed into triangles DEFD E F and DFBD F B. As the line segments DFD F and BHB H are parallel, sliding the apex of triangle DFBD F B from BB along to OO does not change its area - the “height” of the apex above DFD F remains constant. So triangles DFBD F B and DFOD F O have the same area. Thus the shaded region has the same area as the kite DEFOD E F O. This in turn is one sixth of the area of the full dodecagon, so when combined with the lower shaded region then the shaded region is one third of the full dodecagon.

To find the area of the dodecagon, consider the triangles AOKA O K and AOLA O L. The first of these, AOKA O K, is equilateral and so AKA K has length 66. Then since the line segment AKA K cuts OLO L perpendicularly, triangle AOLA O L has base OLO L of length 66 and height half of AKA K of length 33, so has area 99. This is one twelfth of the area of the whole polygon, which therefore has area 108108.

The shaded region then has area 3636.