# Learntofish's Blog

## How to draw regular polygons

Posted by Ed on November 20, 2012

In this blog post we will derive a formula for the interior angle $\alpha$ of an n-sided regular polygon. We will examine how $\alpha$ depends on n. This allows us to draw the regular polygon.

Regular polygons
A regular polygon is a figure whose sides have the same length (equilateral) and whose interior angles are equal (equiangular). Let’s draw one with 3 sides. We can easily construct it if we know the interior angle $\alpha$. For the triangle it is $\alpha$ = 60°.

Let’s do the same for a square, for which the interior angle is $\alpha$ = 90°.

Now, can we do the same for a pentagon (5 sides)? What about a 13 sided regular polygon?

Obviously, we could if we knew how to choose $\alpha$. Before you continue reading you may want to ponder how to determine the internal angle. So, spoilers ahead.

Derivation of interior angle

We have two equations.
$\alpha + \beta = 180^\circ$ (1)
$n \cdot \beta = 360^\circ$ (2)

We solve (2) for $\beta$: $\beta = \frac{360^\circ}{n}$ (3)

Plug (3) into (1): $\alpha + \frac{360^\circ}{n} = 180^\circ$ (4)

Multiply (4) by n: $n \alpha + 360^\circ = n \cdot 180^\circ$ (5)

Substract 2*180° on both sides: $n \alpha = n \cdot 180^\circ - 2 \cdot 180^\circ = (n-2) \cdot 180^\circ$ (6)

Divide by n: $\alpha = \frac{(n-2)}{n} \cdot 180^\circ$ (7)

This is the interior angle for an n-sided regular polygon.

Problems

1) Show that the formula is correct for the triangle and the square.

2) What is the interior angle for the pentagon?

3) What is the sum of all interior angles in a 7-sided regular polygon?

4) What is the sum of all interior angles in an n-sided regular polygon?

5) What happens with the interior angle if you make n bigger and bigger?

1) Before we derived the formula we already knew that the interior angles are 60° and 90° respectively for the triangle and square. Let’s check if the formula gives us these values.

For the triangle we set n=3 and get $\alpha=60^\circ$.
For the square we set n=4 and get $\alpha=90^\circ$.

2) For the pentagon we set n=4 and get $\alpha=108^\circ$.

3) and 4) The sum of all interior angles in an n-sided regular polygon is $n\alpha = (n-2)\cdot 180^\circ$. For a 7-sided regular polygon we set n=7 and get $7\alpha = (7-2)\cdot 180^\circ = 5 \cdot 180^\circ = 900^\circ$.

5) If we make n bigger and bigger, the value for $\alpha$ approaches 180°. We can express this with the limit:

$\lim_{n\rightarrow \infty} \alpha$
$= \lim_{n\rightarrow \infty} \frac{(n-2)}{n} \cdot 180^\circ$
$= \lim_{n\rightarrow \infty} \left(1-\frac{2}{n}\right) \cdot 180^\circ$
$= 180^\circ$