Angles in polygons

We are going to look at the properties of angles in regular and irregular polygons.

The diagram shows a regular pentagon with a bold outline. The sides of the shape are extended. The blue angles, inside the shape, are called internal angles and the red angles, outside of it, are called external angles.

Here is an irregular pentagon. The irregular pentagon also has internal and external angles. I’ve marked one of each in the diagram.

The internal and external angles of the regular and irregular polygons lie on a straight line. This means that they sum to $180^circ$ .

All of the external angles of all polygons, regular or irregular sum to $360^\circ$ . We can show this geometrically.

If we pick up and move each side of the polygon

And arrange each side so that they all start from the same point

In the final image, all of the sides of the polygon start from the same point and it is clear that the angles form a circle around the point and so the angles add to $360^\circ$ .

We can see that this is also true for irregular polygons.

For regular polygons only, the exterior angles are all equal. We can calculate the external angle by dividing $360^\circ$ by the number of external angles. The number of external angles is the same as the number of sides. For regular polygons, therefore, the external angles are equal to $\frac{360^\circ}{number of sides}$ . For a pentagon this is $\frac{360^\circ}{5} = 72^\circ$ . This means that the internal angles are $180 - 72 = 108^\circ$

Previous Lesson

Back to Course

Next Lesson

Trigonometry

Angles in polygons