# Equilateral Areas

## Problem

*ABC* and *DEF* are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of *ABC* and *DEF* .

Does this work for any whole number side lengths?

If not, under what circumstances does it work?

What if the lengths of the sides of the triangles had been *a* and *b* instead of 3 and 4?

Can you construct an equilateral triangle whose area is the sum of the areas of *ABC* and *DEF*? What is the new area?

## Getting Started

Drawing the triangles on isometric paper and using areas bsed on triangles rather than squares might help.

## Student Solutions

John wrote:

Areas of triangles using triangluar measure generate the square numbers

$1, 4, 9, 16, 25$.

So the two triangles $3$ and $4$ were a fairly special case as $3^2 + 4^2 = 5^2$

But there are others that work such as $5, 12,13$ - that is Pythagorean Triples.

In the original problem $a = 3$ and $b = 4$, so $3^2 + 4^2 = c^2$ giving $c = 5$.

This was essentially just another way of looking at Pythagoras's theorem.

In general:

The formula for the area of an equilateral triangle with side $x$ is

$\text{Area} = \frac{x^2\sqrt3}{4}$So with the two triangles with sides *a* and *b* respectively, we are looking for a third triangle with area:

$$\frac{c^2\sqrt3}{4} = \frac{a^2\sqrt3}{4} + \frac{b^2\sqrt3}{4} $$