Surprise, surprise, a Catriona Shearer puzzle.
Going, Going, ‘gon
Six identical squares and a smaller rectangle are fitted into this regular hexagon. What fraction of the hexagon do they cover?
Is there any way to do this besides the hard way? I looked for a dissection solution but came up with nothing.
Look at just one twelfth of the hexagon, a right triangle, :
and are two sides of a square. and are two line segments I added. The central rectangle isn’t shown here because it differs in the different triangles around the hexagon.
If is defined to be length 1 and then and . But and , so
is the correct solution
Then the height of the triangle is and its area is , and the hexagon’s full area is .
The sides of the squares are , so the six squares have area .
The central rectangle has area .
Then the squares plus rectangle are , which is of the hexagon’s area.