## Perimeter pickle

By David Shaw, 2 October 2019

**Difficulty: Tricky**

Which shape will give Imogen’s goats more area?

Imogen has had enough of her goats getting into her veggie patch. She’s bought wire and poles, but before she starts building, she must work out what shape fence she needs.

The two options she’s tossing up are either an equilateral triangle with two metre sides, or a regular hexagon with one metre sides. Either way, the perimeter will be the same: six metres.

Imogen wants the option with more area inside. Can you help her work out which option that is?

**Scroll down or click for a hint, or the answer!**

## Brainteaser hint

It’s tricky to work out the area inside a hexagon – can you cut it into triangles?

## Brainteaser answer

Since the hexagon fence contains more one metre triangles, it has the bigger area!

Let’s start with a bit of intuition. Round shapes are better at holding big areas than spiky ones – that’s why bubbles are round. A regular hexagon is rounder than a triangle, so it’s probably going to have more area inside. But to prove it, you’ll need to use maths.

Hexagons and triangles are different shapes, which makes them hard to compare. Luckily, you can turn the hexagon into equilateral triangles quite easily. If you connect opposite corners, it’s like six pizza slices, each an equilateral triangle with one metre sides.

How about the triangular fence? Each of its sides are two metres long, so cut it into one metre triangles. Find the midpoint of each side and connect them, cutting the big triangle into four triangles, each with one metre sides.

Since the hexagon fence contains more one metre triangles, it has the bigger area!

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