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Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

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LOGO Challenge 6 - Triangles and Stars

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.

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Hexagon Cut Out

Weekly Problem 52 - 2012
An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?

Triangle in a Hexagon

Stage: 3 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Call the length of one side of the hexagon $s$ and the height of the hexagon $h$:

hexagon with height and side marked

So the area of the shaded triangle is $$\frac{1}{2} \times s \times h$$

Now divide the hexagon into $6$ equilateral triangles:

Hexagon divided into 6 equilateral triangles

Each triangle has area $$\frac{1}{2}\times s \times \frac{h}{2}$$ so the area of the hexagon is $$6 \times \frac{1}{2}\times s \times \frac{h}{2}=3 \times \frac{1}{2} \times s \times h$$ or $$3 \times \text{Shaded area}$$

So the shaded area is $\frac{1}{3}$ of the area of the whole hexagon.

This problem is taken from the UKMT Mathematical Challenges.