# Ratio of areas

What is the ratio of the area of the hexagon to the area of the triangle?

What is the ratio of the area of a regular hexagon with side length 1 to that of an equilateral triangle of side length 4?

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*This problem is taken from the World Mathematics Championships*

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If you draw lines from the centre of the hexagon to each of its vertices, 6 identical isosceles triangles are formed (with the green angles equal).

The blue angle is 360$^\circ\div$6 = 60$^\circ$, which means that the green angles must also be 60$^\circ$, and so the hexagon is split into 6 equilateral triangles - which have side length 1, the same as the hexagon.

**Splitting both shapes into equilateral triangles with side length 1**

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Joining the midpoints of the sides, as shown on the right, we can split the triangle into 4 equilateral triangles with side length 2.

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Doing the same to each of the smaller triangles formed gives equilateral triangles with side length 1. There are 4$\times$4 = 16 of these in the large equilateral triangle.

So the hexagon contains 6 equilateral triangles of side length 1, whilst the triangle contains 16. So their areas are in the ratio 6:16, which simplifies to 3:8.

6 copies of the triangle will fit together to make a hexagon with side length 4.

So 6 copies of the triangle will make a copy of the hexagon which has been enlarged by a scale factor of 4.

That means its area has been enlarged by a scale factor of 4$^2$ = 16.

So 6 copies of the triangle have the same area as 16 copies of the hexagon.

So the ratio of the area of the hexagon to the area of the triangle is 6:16, which simplifies to 3:8

So the hexagon contains 6 equilateral triangles of side length 1, whilst the triangle contains 16. So their areas are in the ratio 6:16, which simplifies to 3:8.

**Scaling the hexagon up**Image

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So 6 copies of the triangle will make a copy of the hexagon which has been enlarged by a scale factor of 4.

That means its area has been enlarged by a scale factor of 4$^2$ = 16.

So 6 copies of the triangle have the same area as 16 copies of the hexagon.

So the ratio of the area of the hexagon to the area of the triangle is 6:16, which simplifies to 3:8

**Scaling the triangle down**

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This means that their area is 4$^2$ = 16 times smaller.

So, when scaled down by a factor of 16, the area of the triangle fits 6 times into the area of the hexagon. So the area of the hexagon is $\frac6{16}=\frac38$ of the area of the triangle.

So the ratio of the area of the hexagon to the area of the triangle is $\frac38$:1, which simplifies to 3:8.