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The term ‘**poly**’ means **many**, so a polygon is just a shape with **many** **sides**. When we say ‘** many**’, we mean

**three**

**or**

**more**. So essentially, a polygon can be any

**2D**

**shape**that's

**not**a

**circle**. A polygon is a

**regular**polygon if all

**sides**and

**Angles**are the

**same**.

## Interior Angles in Polygons

When we talk about what angles add up to a polygon, we refer to the **sum of interior angles**. We will use this term a lot from now on, so it is essential to know it.

For a polygon, an **interior angle** is an angle inside the polygon (see the diagram above). The **sum** **of** **interior** **angles** is what all of the angles inside the polygon **add** **up** **to**. So, formally, we already know that the sum of interior angles in a triangle is $180\xb0$ and in a quadrilateral is $360\xb0$.

### Sum of Interior Angles Formula

Previously, we have just been expected to know that the interior angles in a triangle sum to $180\xb0$ and the interior angles in a quadrilateral sum to $360\xb0$. We have just taken it as a fact and have never really questioned it. However, you may now be thinking, *why*** **is this the case? Or you may not... However, a convenient formula tells us the sum of interior angles for any polygon. It goes as follows...

For any given polygon with n sides,

$SumofInteriorAngles=(n-2)\times {180}^{\xb0}$

So, when we have a triangle, $n=3$ and so the sum of interior angles is $(3-2)\times 180=180\xb0$.

Similarly, when we have a quadrilateral, $n=4$ and so the sum of interior angles is $(4-2)\times 180=360\xb0$

We already knew those two results. However, now we can apply this formula to shapes with more than four sides.

**Calculate the sum of interior angles for a pentagon. **

**Solution:**

A pentagon has five sides, so using the formula, the sum of interior angles is $(5-2)\times 180=540\xb0$

**Calculate the sum of interior angles for a nonagon.**

**Solution:**

A nonagon has nine sides, so using the formula, the sum of interior angles is $(9-2)\times 180=1260\xb0$

**Calculate the sum of interior angles for the shape below.**

**Solution:**

The above shape has 14 sides and so the sum of interior angles is $(14-2)\times 180=2160\xb0$

**Calculate the sum of interior angles for a 24 sided shape. **

**Solution:**

When id="2555885" role="math" $n=24$, the sum of interior angles is $(24-2)\times 180=3960\xb0$

**Calculate the size of the angle x in the image below. **

**Solution: **

This shape has five sides, so the sum of interior angles is $(5-2)\times 180=540\xb0$

Each of the right angles in the shape is $90\xb0$and so we can work out the missing angle by subtracting all of the given angles from 540. Thus, $x=540-90-90-90-130=140\xb0$

#### Table of Common Interior Angles

The below table shows the sum of interior angles for the first eight polygons. However, you could confirm these results for yourself using the formula.

Shape | # sides | Sum of Interior Angles ($\xb0$) |

Triangle | 3 | 180 |

Quadrilateral | 4 | 360 |

Pentagon | 5 | 540 |

Hexagon | 6 | 720 |

Heptagon | 7 | 900 |

Octagon | 8 | 1080 |

Nonagon | 9 | 1260 |

Decagon | 10 | 1440 |

#### Calculating Each Interior Angle

Earlier, we defined regular polygons as polygons with **equal** **sides** and **angles**. We therefore may wish to calculate **each** **interior** **angle** of a regular polygon. We first calculate the **sum** **of** **interior** **angles** and **divide** this Number by the **Number of sides**.

**Calculate each interior angle for a regular hexagon.**

**Solution: **

Using table 1, we can see that the sum of interior angles for a hexagon is $720\xb0$. Since this hexagon is regular, each of the angles is the same and thus we can work out each interior angle by dividing $720$ by $6$. Therefore, each interior angle is $120\xb0$.

**Below is part of a tiling pattern consisting of three regular pentagons. Calculate the angle labelled x. **

**Solution: **

The sum of interior angles for each regular hexagon is $720\xb0$ (using the table of common interior angles).

Thus, each interior angle in each hexagon is $120\xb0$.

Recall that angles around a point sum to 360 degrees. Therefore, x can be found by subtracting the other known angles from 360. Thus, $x=360-108-108=144\xb0$

## Exterior Angles in Polygons

There is also an exterior angle for each interior angle in a polygon. An exterior angle is formed between any **side** of the **shape** and the **straight** **line** **extended** outside of the shape. This may sound not very clear, but it is easier to see illustrated.

In the diagram above, the **interior** angles are labelled orange, and the exterior angles are **green**. Since the exterior angle lies on the **same** **straight** **line** as the interior angle, the **sum of the interior and exterior angles is $180\xb0$.** Therefore, an **exterior** angle can be calculated by **subtracting** the **interior** angle from $180\xb0$.

**In the image below, angles x and y are exterior angles. Calculate x and y.**

**Solution: **

For exterior angle x, the interior angle is $109\xb0$. Thus, since angles on a straight line add up to $180\xb0$, $x=180-109=71\xb0$. Angle $y$ is another exterior angle and since angles on a straight line add to 180, $y=180-81=99\xb0$.

**Calculate each exterior angle of a regular heptagon.**A heptagon has seven sides and thus the sum of interior angles is $900\xb0$Since this heptagon is regular, we can work out each interior angle by dividing 900 by 7 to obtain $128.6\xb0$. Therefore, we can calculate each exterior angle by subtracting this from 180. Thus, each exterior angle is $180-128.6=51.4\xb0$.

**Solution:**A heptagon is also sometimes referred to as a septagon.

### Sum of Exterior Angles

The **sum** of **exterior** **angles** for any polygon is dead simple. It is $360\xb0$. Unlike interior angles, we do not need to memorise any fancy formulae to work out the sum of exterior angles; we simply need to remember the sum of exterior angles for any polygon $360\xb0$. Using this, we can begin to answer some more questions.

**Each exterior angle of a regular polygon is 10. Work out the number of sides the polygon has. **

**Solution: **

Since the sum of exterior angles is $360\xb0$, and each exterior angle is $10\xb0$, we can calculate the number of sides by $360\xf710=36$. Thus, this polygon has **36 sides.**

**Each interior angle of a regular polygon is 165. Work out the number of sides the polygon has. **

**Solution: **

If each interior angle is $165$, each exterior angle must be $180-165=15\xb0$. Since the sum of the exterior angles is $360\xb0$, there must be $360\xf715=24$** sides.**

## Angles in polygons - Key takeaways

- The interior angles in a polygon are the angles inside the polygon.
- To calculate the sum of interior angles, subtract two from the number of sides and multiply the result by 180 degrees.
- If the polygon is regular, each of the sides is the same.
- An exterior angle is formed between any side of the shape and the straight line extended outside of the shape.
- The sum of exterior angles of any polygon is 360 degrees, regardless of the number of sides.

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##### Frequently Asked Questions about Angles in Polygons

What do angles in a polygon add up to?

It is different for each polygon. The sum of interior angles in a regular polygon can be found by subtracting two from the number of sides and then multiplying this result by 180 degrees.

What is the sum of the exterior angles of a polygon?

The sum of exterior angles is 360 degrees for any polygon.

What is the formula for the sum of the interior angles of a polygon?

(n-2) x 180

What is the sum of the interior angles of a polygon?

The sum of interior angles in a regular polygon can be found by subtracting two from the number of sides and then multiplying this result by 180 degrees.

How to find the missing angle in a polygon?

First work out what the sum of angles should be, and then subtract the angles that you know to work out the missing one.

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