# 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?

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?

## Problem

Image

The diagram shows an irregular hexagon with interior angles all equal to 120 degrees made by cutting the corners off a piece of card in the shape of an equilateral triangle with sides of length 20 units.

An identical hexagon could also be made by cutting the corners off a different equilateral triangle.

What is the side length of this triangle?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

The other triangle has side length 27.

By extending the short sides, one can construct another triangle. Since all the angles involved are 60 degrees, the new small triangles and the new large triangle are all equilateral. The sides of the new, large triangle are:

13 + 3 + 11 = 11 + 6 + 10 = 10 + 4 + 13 = 27