Interior Angle Sum Theorem
What is true about the sum of interior angles of a polygon ?
The sum of the measures of the interior angles of a convex polygon with n sides is $ (n-2)\cdot180^{\circ} $
Shape | Formula | Sum Interior Angles |
---|---|---|
$$ \red 3 $$ sided polygon
(triangle) |
$$ (\red 3-2) \cdot180 $$ | $$ 180^{\circ} $$ |
$$ \red 4 $$ sided polygon
(quadrilateral) |
$$ (\red 4-2) \cdot 180 $$ | $$ 360^{\circ} $$ |
$$ \red 6 $$ sided polygon
(hexagon) |
$$ (\red 6-2) \cdot 180 $$ | $$ 720^{\circ} $$ |
Problem 1
![triangle](images/triangle.gif)
180°
You can also use Interior Angle Theorem:$$ (\red 3 -2) \cdot 180^{\circ} = (1) \cdot 180^{\circ}= 180 ^{\circ} $$
Problem 2
![triangle](images/quadrilateral.gif)
360° since this polygon is really just two triangles and each triangle has 180°
You can also use Interior Angle Theorem:$$ (\red 4 -2) \cdot 180^{\circ} = (2) \cdot 180^{\circ}= 360 ^{\circ} $$
Problem 3
![Polygon Image](images/five-gon.gif)
Use Interior Angle Theorem:$$ (\red 5 -2) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ} $$
Problem 4
![Polygon Image](images/hexagon.png)
Use Interior Angle Theorem: $$ (\red 6 -2) \cdot 180^{\circ} = (4) \cdot 180^{\circ}= 720 ^{\circ} $$
Video Tutorial
on Interior Angles of a Polygon
Definition of a Regular Polygon:
Measure of a Single Interior Angle
Shape | Formula | Sum interior Angles |
---|---|---|
Regular Pentagon | $$ (\red 3-2) \cdot180 $$ | $$ 180^{\circ} $$ |
$$ \red 4 $$ sided polygon
(quadrilateral) |
$$ (\red 4-2) \cdot 180 $$ | $$ 360^{\circ} $$ |
$$ \red 6 $$ sided polygon
(hexagon) |
$$ (\red 6-2) \cdot 180 $$ | $$ 720^{\circ} $$ |
What about when you just want 1 interior angle?
In order to find the measure of a single interior angle of a regular polygon (a polygon with sides of equal length and angles of equal measure) with n sides, we calculate the sum interior anglesor $$ (\red n-2) \cdot 180 $$ and then divide that sum by the number of sides or $$ \red n$$.
The Formula$ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} $
Example 1
Let's look at an example you're probably familiar with-- the good old triangle $$\triangle$$ . Now, remember this new rule above only applies to regular polygons. So, the only type of triangle we could be talking about is an equilateral one like the one pictured below.![Equilateral triangle picture](https://www.mathwarehouse.com/geometry/triangles/images/equilateral-triangles/equilateral-triangle.png)
You might already know that the sum of the interior angles of a triangle measures $$ 180^{\circ}$$ and that in the special case of an equilateral triangle, each angle measures exactly $$ 60^{\circ}$$.
![Equilateral triangle picture angle labelled](https://www.mathwarehouse.com/geometry/triangles/images/equilateral-triangles/equilateral-triangle-60-degrees.png)
So, our new formula for finding the measure of an angle in a regular polygon is consistent with the rules for angles of triangles that we have known from past lessons.
Example 2
To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: $ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} \\ \frac{(\red8-2) \cdot 180}{ \red 8} = 135^{\circ} $
Finding 1 interior angle of a regular Polygon
Problem 5
Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle
![poly1](img-equations/single-angle-1.gif)
Problem 6
Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle
![poly2](img-equations/single-angle-2.gif)
Problem 7
Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle
![poly2](img-equations/single-angle-3.gif)
Challenge Problem
This question cannot be answered because the shape is not a regular polygon. You can only use the formula to find a single interior angle if the polygon is regular!
Consider, for instance, the irregular pentagon below.
You can tell, just by looking at the picture, that $$ \angle A and \angle B $$ are not congruent.
![pentagon irregular](images/irregular-pentagon-INTERIOR-B.gif)
The moral of this story- While you can use our formula to find the sum of the interior angles of any polygon (regular or not), you can not use this page's formula for a single angle measure--except when the polygon is regular.
How about the measure of an exterior angle?
Formula for sum of exterior angles:
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.
Measure of a Single Exterior Angle
Formula to find 1 angle of a regular convex polygon of n sides =
![Exterior Angles of Triangle](images/triangle-exterior.png)
$$ \angle1 + \angle2 + \angle3 = 360° $$
![Exterior Angles of Polygon](images/quadrilateral-exterior.png)
$$ \angle1 + \angle2 + \angle3 + \angle4 = 360° $$
![Exterior Angles of Pentagon](images/pentagon-extior.png)
$$ \angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360° $$
Practice Problems
Problem 8
Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle
![poly2](img-equations/exterior-angles/problem1.gif)
Problem 9
Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle
![poly2](img-equations/exterior-angles/problem2.gif)
Problem 10
Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle
![poly2](img-equations/exterior-angles/problem3.gif)
Challenge Problem
This question cannot be answered because the shape is not a regular polygon. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular!
Consider, for instance, the pentagon pictured below. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$ \angle A \text{ and } and \angle B $$ are not congruent..
![](images/irregular-pentagon.gif)
Determine Number of Sides from Angles
It's possible to figure out how many sides a polygon has based on how many degrees are in its exterior or interior angles.
Problem 11
Use formula to find a single exterior angle in reverse and solve for 'n'.
![equation](img-equations/mixed/problem1.gif)
Problem 12
Use formula to find a single exterior angle in reverse and solve for 'n'.
![equation](img-equations/mixed/problem2.gif)
Problem 13
Use formula to find a single exterior angle in reverse and solve for 'n'.
![equation](img-equations/mixed/problem4.gif)
Challenge Problem
When you use formula to find a single exterior angle to solve for the number of sides , you get a decimal (4.5), which is impossible. Think about it: How could a polygon have 4.5 sides? A quadrilateral has 4 sides. A pentagon has 5 sides.
![challenge](img-equations/mixed/problem5.gif)