Sum of Interior & Exterior Angles

Written by

Malcolm McKinsey

January 26, 2023

Fact-checked by

Paul Mazzola

Finding the sum of interior & exterior angles

Polygons are like the little houses of two-dimensional geometry world. They create insides, called the interior, and outsides, called the exterior. You can measure interior angles and exterior angles. You can also add up the sums of all interior angles, and the sums of all exterior angles, of regular polygons.

Our formula works on triangles, squares, pentagons, hexagons, quadrilaterals, octagons and more.

What is a regular polygon?

For a polygon to be a regular polygon, it must fulfill these four requirements:

• Be two-dimensional

• Enclose a space, creating an interior and exterior

• Use only line segments for sides

• Have all sides equal in length to one another, and all interior angles equal in measure to one another

Sum of interior angles of a polygon

Regular polygons exist without limit (theoretically), but as you get more and more sides, the polygon looks more and more like a circle. The regular polygon with the fewest sides – three – is the equilateral triangle.

The regular polygon with the most sides commonly used in geometry classes is probably the dodecagon, or 12-gon, with 12 sides and 12 interior angles:

Pretty fancy, isn't it? But just because it has all those sides and interior angles, do not think you cannot figure out a lot about our dodecagon. Suppose, for instance, you want to know what all those interior angles add up to, in degrees?

Sum of interior angles

Triangles are easy. Their interior angles add to 180°. Likewise, a square (a regular quadrilateral) adds to 360° because a square can be divided into two triangles.

The word "polygon" means "many angles," though most people seem to notice the sides more than they notice the angles, so they created words like "quadrilateral," which means "four sides."

Regular polygons have as many interior angles as they have sides, so the triangle has three sides and three interior angles. Square? Four of each. Pentagon? Five, and so on.

Our dodecagon has 12 sides and 12 interior angles.

Sum of interior angles formula

The formula for the sum of that polygon's interior angles is refreshingly simple. Let n equal the number of sides of whatever regular polygon you are studying. Here is the formula:

Sum of angles in a triangle

You can do this. Try it first with our equilateral triangle:

Sum of angles of a square

And again, try it for the square:

How to find one interior angle

To find the measure of a single interior angle, then, you simply take that total for all the angles and divide it by n, the number of sides or angles in the regular polygon.

The new formula looks very much like the old formula:

Again, test it for the equilateral triangle:

And for the square:

Hey! It works! And it works every time. Let's tackle that dodecagon now.

Interior angles examples

Remember what the 12-sided dodecagon looks like? Let's find the sum of the interior angles, as well as one interior angle:

Find the sum of interior angles of a dodecagon

Now, let's find one interior angle

Sum of exterior angles

Every regular polygon has exterior angles. These are not the reflex angle (greater than 180°) created by rotating from the exterior of one side to the next. That is a common misunderstanding.

For instance, in an equilateral triangle, the exterior angle is not 360° - 60° = 300°, as if we were rotating from one side all the way around the vertex to the other side.

Since you are extending a side of the polygon, that exterior angle must necessarily be supplementary to the polygon's interior angle. Together, the adjacent interior and exterior angles will add to 180°.

For our equilateral triangle, the exterior angle of any vertex is 120°. For a square, the exterior angle is 90°.

Exterior angle formula

If you prefer a formula, subtract the interior angle from 180°:

Exterior angles examples

What do we have left in our collection of regular polygons? That dodecagon! We know any interior angle is 150°, so the exterior angle is:

Checking your work

Look carefully at the three exterior angles we used in our examples:

Prepare to be amazed. Multiply each of those measurements times the number of sides of the regular polygon:

• Triangle = 120°×3=360°

• Square = 90°×4=360°

• Dodecagon = 30°×12 =360°

It looks like magic, but the geometric reason for this is actually simple: to move around these shapes, you are making one complete rotation, or turn, of 360°.

Still, this is an easy idea to remember: no matter how fussy and multi-sided the regular polygon gets, the sum of its exterior angles is always 360°.

Lesson summary

After working through all that, now you are able to define a regular polygon, measure one interior angle of any polygon, and identify and apply the formula used to find the sum of interior angles of a regular polygon.

You also can explain to someone else how to find the measure of the exterior angles of a regular polygon, and you know the sum of exterior angles of every regular polygon.