Instructor: Malcolm M.

You are watching: How to find how many sides a polygon has

Malcolm has actually a Master"s degree in education and holds 4 teaching certificates. He has been a public college teacher because that 27 years, including 15 years as a math teacher.

Interior edge Formula (Definition, Examples, amount of inner Angles)

VideoDefinitionSum of interior AnglesFinding Unknown AnglesRegular Polygons

If you take it a watch at other geometry great on this valuable site, you will see that we have actually been mindful to cite interior angles, not simply angles, when stating polygons. Every polygon has actually interior angles and also exterior angles, however the inner angles space where every the interesting action is.


What you"ll learn:

After functioning your means through this lesson and video, you will be able to:

Identify inner angles that polygonsRecall and also apply the formula to uncover the amount of the interior angles that a polygonRecall a an approach for recognize an unknown internal angle of a polygonCalculate internal angles the polygonsDiscover the variety of sides that a polygon

Interior angle Formula

From the easiest polygon, a triangle, to the infinitely complicated polygon through n sides, sides of polygon close in a space. Every intersection of sides creates a vertex, and also that vertex has actually an interior and also exterior angle. Interior angles of polygons space within the polygon.

Though Euclid did sell an exterior angle theorem specific to triangles, no interior Angle organize exists. Instead, you deserve to use a formula the mathematically describes an exciting pattern about polygons and also their inner angles.

Sum of internal Angles Formula

This formula permits you come mathematically divide any type of polygon into its minimum number of triangles. Because every triangle has actually interior angles measuring 180°, multiplying the number of dividing triangles time 180° offers you the amount of the inner angles.


S = (n - 2) × 180°

S = sum of interior angles

n = number of sides of the polygon

Try the formula ~ above a triangle:

S = (n - 2) × 180°

S = (3 - 2) × 180°

S = 1 × 180°

S = 180°

Well, the worked, however what around a more complex shape, like a dodecagon?

It has 12 sides, so:

S = (n - 2) × 180°

S = (12 - 2) × 180°

S = 10 × 180°

S = 1,800°

How perform you know that is correct? Take any kind of dodecagon and also pick one vertex. Connect every various other vertex to the one with a straightedge, separating the an are into 10 triangles. Ten triangles, every 180°, renders a full of 1,800°!

Finding an Unknown internal Angle

The very same formula, S = (n - 2) × 180°, can help you find a missing interior angle of a polygon. Right here is a strange pentagon, through no 2 sides equal:

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

S = (n - 2) × 180°

S = (5 - 2) × 180°

S = 3 × 180°

S = 540°

So subtracting the 4 known angle from 540° will certainly leave you v the missing angle:

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

The unknown edge is 100°.

Finding inner Angles of regular Polygons

Once you know how to discover the amount of interior angles of a polygon, recognize one interior angle for any regular polygon is just a matter of dividing.

Where S = the sum of the inner angles and n = the number of congruent political parties of a continual polygon, the formula is:


Here is one octagon (eight sides, eight inner angles). First, usage the formula for finding the amount of internal angles:

S = (n - 2) × 180°

S = (8 - 2) × 180°

S = 6 × 180°

S = 1,080°

Next, division that sum by the variety of sides:

measure that each internal angle = Snmeasure the each internal angle = 1,080°8measure that each internal angle = = 135°

Each inner angle that a continual octagon is = 135°.

Finding the number of Sides of a Polygon

You can use the same formula, S = (n - 2) × 180°, to uncover out how numerous sides n a polygon has, if you know the value of S, the amount of internal angles.

You understand the amount of internal angles is 900°, however you have no idea what the form is. Use what you understand in the formula to discover what you execute not know:

State the formula:

S = (n - 2) × 180°

Use what you know, S = 900°

900° = (n - 2) × 180°

Divide both political parties by 180°

900°180° = ((n - 2) × 180°)180°

No require 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 also you can recall and also apply the formula, S = (n - 2) × 180°, to find the sum of the internal angles that a polygon. You additionally are able to recall a method for recognize an unknown inner angle that a polygon, by subtracting the known interior angles native the calculate sum.

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Not only all that, but you can also calculate interior angles of polygons making use of Sn, and also you can find the variety of sides of a polygon if you understand the amount of their inner angles. The is a entirety lot the knowledge collected from one formula, S = (n - 2) × 180°.