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.

- Video
- What Is A Regular Polygon?
- Sum of Interior Angles of a Polygon
- Sum of Interior Angles
- Sum of Exterior Angles

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

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?

Triangles are easy. Their **interior angles** add to $180\xb0$. Likewise, a square (a regular quadrilateral) adds to $360\xb0$ 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.

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:

$Sumofinteriorangles=(n-2)\times 180\xb0$

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

$(n-2)\times 180\xb0$

$(3-2)\times 180\xb0$

Sum of interior angles = $180\xb0$

And again, try it for the square:

$(n-2)\times 180\xb0$

$(4-2)\times 180\xb0$

$2\times 180\xb0$

Sum of interior angles = $360\xb0$

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:

$Oneinteriorangle=\frac{(n-2)\times 180\xb0}{n}$

Again, test it for the equilateral triangle:

$\frac{(3-2)\times 180\xb0}{3}$

$\frac{180\xb0}{3}$

One interior angle = $60\xb0$

And for the square:

$\frac{(4-2)\times 180\xb0}{4}$

$\frac{2\times 180\xb0}{4}$

$\frac{360\xb0}{4}$

One interior angle = $90\xb0$

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

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

$(n-2)\times 180\xb0$

$(12-2)\times 180\xb0$

$10\times 180\xb0$

Sum of interior angles = $\mathrm{1,800}\xb0$

$\frac{(n-2)\times 180\xb0}{n}$

$\frac{(12-2)\times 180\xb0}{12}$

$\frac{10\times 180\xb0}{12}$

$\frac{\mathrm{1,800}\xb0}{12}$

One interior angle = $150\xb0$

Awesome!

Every regular polygon has **exterior angles**. These are *not* the reflex angle (greater than $180\xb0$) 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.

Exterior angles are created by extending one side of the regular polygon past the shape, and then measuring in degrees from that extended line back to the next side of the polygon.

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 $180\xb0$.

For our equilateral triangle, the exterior angle of any vertex is $120\xb0$. For a square, the exterior angle is $90\xb0$.

If you prefer a formula, subtract the interior angle from $180\xb0$:

$Exteriorangle=180\xb0-interiorangle$

What do we have left in our collection of regular polygons? That dodecagon! We know any interior angle is $150\xb0$, so the exterior angle is:

$180\xb0-150\xb0$

Exterior angle = $30\xb0$

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

Triangle = $120\xb0$

Square = $90\xb0$

Dodecagon = $30\xb0$

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

- Triangle = $120\xb0\times 3=360\xb0$
- Square = $90\xb0\times 4=360\xb0$
- Dodecagon = $30\xb0\times 12=360\xb0$

Every time you add up (or multiply, which is fast addition) the sums of exterior angles of any regular polygon, you *always* get 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°**.

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.

After working your way through this lesson and the video, you learned to:

- Define a regular polygon
- Identify and apply the formula used to find the sum of interior angles of a regular polygon
- Measure one interior angle of a polygon using that same formula
- Explain how you find the measure of any exterior angle of a regular polygon
- Know the sum of the exterior angles of every regular polygon

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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