Ratios of the Areas of an Internal Hexagon to Its Star to the External Hexagon
This Demonstration finds the ratios of the areas of an internal hexagon to its star to the external hexagon without doing calculations.
When the "show step 1" box is checked, you can move the slider below the box to map the small triangles that form the six-pointed star to the internal hexagon. The area of the six-pointed star is equal to twice the area of the hexagon.
Once "show step 2" is checked, you can move the slider below the box to map the triangles into the internal part of the hexagon.
After checking the "show step 3," box, the other three external triangles that complete the external hexagon fill the leftover space in the internal hexagon when you move the slider below that box. The area of the external hexagon is equal to three times the area of the internal hexagon.