Interior Angle Formula

Interior Angle Formula (Definition, Examples, Sum of Interior Angles)

Video Definition Sum of Interior Angles Finding Unknown Angles Regular Polygons

If you take a look at other geometry lessons on this helpful site, you will see that we have been careful to mention interior angles, not just angles, when discussing polygons. Every polygon has interior angles and exterior angles, but the interior angles are where all the interesting action is.

Interior Angle Formula

From the simplest polygon, a triangle, to the infinitely complex polygon with $n$ sides, sides of polygons close in a space. Every intersection of sides creates a vertex, and that vertex has an interior and exterior angle. Interior angles of polygons are within the polygon.

Though Euclid did offer an exterior angles theorem specific to triangles, no Interior Angle Theorem exists. Instead, you can use a formula that mathematically describes an interesting pattern about polygons and their interior angles.

Sum of Interior Angles Formula

This formula allows you to mathematically divide any polygon into its minimum number of triangles. Since every triangle has interior angles measuring $180°$, multiplying the number of dividing triangles times $180°$ gives you the sum of the interior angles.

$S = sum of interior angles$

$n = number of sides of the polygon$

Try the formula on a triangle:

$S = (n - 2) \times 180°$

$S = (3 - 2) \times 180°$

$S = 1 \times 180°$

$S = 180°$

Well, that worked, but what about a more complicated shape, like a dodecagon?

[insert dodecagon drawing]

It has 12 sides, so:

$S = (n - 2) \times 180°$

$S = (12 - 2) \times 180°$

$S = 10 \times 180°$

$S = \mathrm{1,800}°$

How do you know that is correct? Take any dodecagon and pick one vertex. Connect every other vertex to that one with a straightedge, dividing the space into 10 triangles. Ten triangles, each $180°$, makes a total of $\mathrm{1,800}°$!

Finding an Unknown Interior Angle

The same formula, $S = (n - 2) \times 180°$, can help you find a missing interior angle of a polygon. Here is a wacky pentagon, with no two sides equal:

[insert drawing of pentagon with four interior angles labeled and measuring 105°, 115°, 109°, 111°; length of sides immaterial]

The formula tells us that a pentagon, no matter its shape, must have interior angles adding to $540°$:

$S = (n - 2) \times 180°$

$S = (5 - 2) \times 180°$

$S = 3 \times 180°$

$S = 540°$

So subtracting the four known angles from $540°$ will leave you with the missing angle:

$540° - 105° - 115° - 109° - 111° = 100°$

The unknown angle is $100°$.

Finding Interior Angles of Regular Polygons

Once you know how to find the sum of interior angles of a polygon, finding one interior angle for any regular polygon is just a matter of dividing.

Where $S$ = the sum of the interior angles and $n$ = the number of congruent sides of a regular polygon, the formula is:

Here is an octagon (eight sides, eight interior angles). First, use the formula for finding the sum of interior angles:

$S = (n - 2) \times 180°$

$S = (8 - 2) \times 180°$

$S = 6 \times 180°$

$S = \mathrm{1,080}°$

Next, divide that sum by the number of sides:

• measure of each interior angle $= \frac{S}{n}$
• measure of each interior angle $= \frac{\mathrm{1,080}°}{8}$
• measure of each interior angle = $= 135°$

Each interior angle of a regular octagon is $= 135°$.

Finding the Number of Sides of a Polygon

You can use the same formula, $S = (n - 2) \times 180°$, to find out how many sides $n$ a polygon has, if you know the value of $S$, the sum of interior angles.

You know the sum of interior angles is $900°$, but you have no idea what the shape is. Use what you know in the formula to find what you do not know:

State the formula:

$S = (n - 2) \times 180°$

Use what you know, $S = 900°$

$900° = (n - 2) \times 180°$

Divide both sides by $180°$

$\frac{900°}{180°} = \frac{\left(\right(n - 2) \times 180°)}{180°}$

No need for parentheses now

$5 = n - 2$

Add 2 to both sides

$5 + 2 = n - 2 + 2$

$7 = n$

The unknown shape was a heptagon!

Lesson Summary

Now you are able to identify interior angles of polygons, and you can recall and apply the formula, $S = (n - 2) \times 180°$, to find the sum of the interior angles of a polygon. You also are able to recall a method for finding an unknown interior angle of a polygon, by subtracting the known interior angles from the calculated sum.

Not only all that, but you can also calculate interior angles of polygons using $\frac{S}{n}$, and you can discover the number of sides of a polygon if you know the sum of their interior angles. That is a whole lot of knowledge built up from one formula, $S = (n - 2) \times 180°$.

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Midsegment of a Triangle