Geometry

Lesson

Interior angles of polygons are all the angles that are inside a shape. Depending on whether we know the number of sides in or the angle sum of a polygon, we can use the information to find:

- the angle sum of the interior angles of the polygon
- the size of each angle in the polygon
- the number of sides in the polygon, or
- the type of polygon.

To do this, we can use a special formula: the total interior angle sum of a polygon is $180\times\left(n-2\right)$180×(`n`−2).

*Proof:*

We've already learned that the angle sum of a triangle is $180^\circ$180°.

Let's consider the following polygon that has been divided up into triangles by drawing lines from one vertex to each of the other vertices:

So, the angle sum of this pentagon is $180^\circ+180^\circ+180^\circ$180°+180°+180°, which is $540^\circ$540°.

Notice that the pentagon has $5$5 sides and it makes $3$3 triangles. The number of triangles produced by drawing lines from one vertex to the others in a polygon, will always be two less than the number of sides in that polygon. This pattern is consistent for all polygons.

This interactive will show you different interior angle sums for different polygons. You can also see how many triangles fit into it.

Watch this video if you would like to see this interactive in action -

Sum of Interior Angles of a Polygon

$\text{Sum of interior angles of a polygon}=180\left(n-2\right)$Sum of interior angles of a polygon=180(`n`−2)

where $n$`n` is the number of sides in the shape

The following examples focus on how to use the formula to find the number of sides in a polygon, and hence how to identify the name of that polygon.

Dave Claims to have drawn a regular polygon with interior angles equal to $100^\circ$100°.

a) Find $n$`n`, the number of sides of such a polygon.

b) What is the name of the shape Dave claims to have drawn?

Deduce the angle properties of intersecting and parallel lines and the angle properties of polygons and apply these properties