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.

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