Surprise, surprise, a Catriona Shearer puzzle.
16.
Going, Going, ‘gon
Six identical squares and a smaller rectangle are fitted into this regular hexagon. What fraction of the hexagon do they cover?
Spoiler!
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.