Copyright © University of Cambridge. All rights reserved.

## 'Polygon Walk' printed from http://nrich.maths.org/

Draw the triangle pointing right such that the rightmost vertex is
at $\mathbf{i}$

The coordinates of a regular $n$-gon with a centred on the origin
with a vertex at $(1,0)$ are

$$\left(\cos\left(\frac{2m\pi}{n}\right),
\sin\left(\frac{2m\pi}{n}\right)\right)\, \text{ where }m=0, \dots,
n-1$$

For a pentagon, the coordinates become

$$

(1, 0), \left(\frac{1}{4}\left(\sqrt{5}-1\right),
\frac{1}{4}\left(\sqrt{10+2\sqrt{5}}\right)\right),
\left(-\frac{1}{4}\left(\sqrt{5}+1\right),
\frac{1}{4}\left(\sqrt{10-2\sqrt{5}}\right)\right)

$$

along with the mirror images in the $x$-axis.

This problem builds on GCSE
vector work and provides a foundation for concepts met in the later
Core A Level modules.