QUICK LINKS


Introduction

QUICK LINKS


Introduction

Interior and Exterior Angles of Polygons

INTRO
The interior angles of a polygon are angles inside the shape. The exterior angles of a polygon are angles outside of the shape formed between any side of the polygon and a line extended from the side next to it.
If we imagine the polygon as a house, the interior angles live inside of the house, while the exterior angles live in exile outside of the house.
We can find the sum of interior angles in a shape with sides using the formula:Just remember that a triangle has , and we can fit triangles in a shape with sides.The sum of exterior angles for all simple polygons is always - see how they form a circle when we combine them?We can use these sums to solve for missing angles inside and outside of polygons.
Check out our
Calculator
or explore our
Lesson
and
Practice
sections to learn more about how to find the area of triangles and test your understanding.

You can also use the Quick Links menu on the left to jump to a section of your choice.

You can also use the Quick Links dropdown above to jump to a section of your choice.

CALCULATOR

Interior & Exterior Angles Calculator

Step 1. Determine whether you’re looking for interior or exterior angles.

If we think of the shape as a house, the interior angles live inside of the house, while the exterior angles live in exile outside of the house.
Pick which type of angles you’re looking for.

Step 2. Calculate the sum of angles.

The sum of exterior angles for all polygons is always .This animation ⬆️ can help you see how the exterior angles combine to create a circle, which is .

Step 3. Find the value of an individual angle.

How many sides does the shape have?
Is the shape regular? This means all the sides are the same length.
We can help you out if you know all but one exterior angle. What are the other -1 angles?
Since the sum of the exterior angles is , we just need to subtract the given angle values to find the unknown angle:
The unknown angle is .
KEY STEPS

How to Find the Interior & Exterior Angles of Polygons

Step 1. Determine whether you’re looking for interior or exterior angles.

If we imagine the polygon as a house, the interior angles live inside of the house, while the exterior angles live in exile outside of the house.Type of Angles:

Step 2. Count the number of sides () the shape has.

Step 3. Calculate the sum of interior angles.

Step 4. Find the value of an individual angle.

If your shape is regular, just divide the sum of the interior angles by the number of sides/angles:
If your shape is irregular and you have the values of all the other interior angles, you can find the missing angle by subtracting your given angles from the sum:

Step 2. Calculate the sum of exterior angles.

The sum of exterior angles for all shapes is always .

Step 3. Find the value of an individual angle.

If your shape is regular, just divide the sum of the exterior angles by the number of sides/angles:
If your shape is irregular and you have the values of all the other exterior angles, you can find the missing angle by subtracting your given angles from the sum:
LESSON
Interior Angles

Interior Angles

The interior angles of a polygon are angles inside the shape formed at the points where the sides connect. If we imagine the polygon as a house, the interior angles live inside the house.We can find the sum of a polygon’s interior angles by using the number of sides the shape has:
To help us remember this formula, we can look at triangles in few different shapes.
We can see that a shape with sides can be divided into a minimum of triangles:
Number of sides:
When we multiply by , we're multiplying the minimum number of triangles that fit in the shape by the degrees in a triangle :

Does this work for all shapes?

This formula works for all shapes, but the triangle trick only works for convex polygons where all interior angles are less than .
Concave polygons have at least one interior angle that is greater than , which breaks the triangle trick.
But the formula will always work!
If your shape is regular, then each side is the same length and each angle is the same size. Check out this table ⬇️ to see a comparison of interior angle values for different shapes:
Regular polygons only
SidesShapeSum of Interior AnglesEach Angle
............
Any Polygon
Let’s take a look at some example problems of how we can use the sum of interior angles to solve for missing angles inside polygons.

EXAMPLE 1

We'll start by finding the value of an interior angle in a regular hexagon (6-sided shape).

Step 1. Determine whether you’re looking for interior or exterior angles.

If we think of the shape as a house, the interior angles live inside of the house, while the exterior angles live in exile outside of the house.
We're looking for the interior angles in this problem.

Step 2. Count the number of sides () the shape has.

Our shape has sides.

Step 3. Calculate the sum of interior angles.

For any convex shape with sides, triangles fit inside:
And since the angles in a triangle always add up to 180°, the sum of the interior angles is equal to.

Step 4. Find the value of an individual angle.

If the shape is regular, we just divide the sum of the interior angles by the number of sides/angles:
The sum of the interior angles in a 6-sided shape is , and each angle is .
Since the sum of the interior angles is , we just need to subtract the given angle values to find the unknown angle:
The unknown angle is .

EXAMPLE 2

Nice work! Now let's try an irregular shape, where the sides and angles are not equal. Find the missing angle in this shape:

Step 1. Determine whether you’re looking for interior or exterior angles.

If we think of the shape as a house, the interior angles live inside of the house, while the exterior angles live in exile outside of the house.
We're looking for the interior angles in this problem.

Step 2. Count the number of sides () the shape has.

Our shape has sides.

Step 3. Calculate the sum of interior angles.

For any convex shape with sides, triangles fit inside:
And since the angles in a triangle always add up to 180°, the sum of the interior angles is equal to.

Step 4. Find the value of an individual angle.

Since the sum of the interior angles is , we just need to subtract the given angle values to find the unknown angle:
The unknown angle is .
Awesome! Expand the
Practice
below to try some more problems or expand the next
Lesson
to explore what we do when we're dealing with exterior angles.
PRACTICE
Interior Angles

Practice: Interior Angles

Question 1 of 6:Find the value of an interior angle in a regular 6-sided shape.

Step 1. Determine whether you’re looking for an interior or exterior angle.

Are we looking for an interior or exterior angle?

Step 2. Count the number of sides (n) on your polygon.

Step 3. Calculate the sum of interior angles.

Step 4. Find the value of the individual angle.

Since we're told the shape is regular, which formula can we use to determine the value of each angle?
Nice! Now let's plug in:

Great work! The value of the missing angle is   !
LESSON
Exterior Angles

Exterior Angles

The exterior angles of a polygon are angles outside of the shape formed between any side of the polygon and a line extended from the next side.
If we imagine the polygon as a house, the exterior angles live in exile outside of the house.
As we decrease the size of any polygon, we can easily see that the exterior angles add up to one full rotation of .
Number of sides: This is true for all polygons, which means the sum of exterior angles for all polygons is always .
Check out this table ⬇️ to see a comparison of exterior angle values for different shapes:
Regular polygons only
SidesShapeSum of Exterior AnglesEach Angle
............
Any Polygon
Let’s take a look at some example problems of how we can use the sum of exterior angles to solve for missing angles outside of polygons.

EXAMPLE 1

We'll start by finding the value of an exterior angle in a regular hexagon (6-sided shape).

Step 1. Determine whether you’re looking for interior or exterior angles.

If we think of the shape as a house, the interior angles live inside of the house, while the exterior angles live in exile outside of the house.
We're looking for the exterior angles in this problem.

Step 2. Calculate the sum of exterior angles.

The sum of exterior angles for all polygons is always .This animation ⬆️ can help you see how the exterior angles combine to create a circle, which is .

Step 3. Find the value of an individual angle.

If the shape is regular, we just divide the sum of the exterior angles by the number of sides/angles:
The sum of the exterior angles in a 6-sided shape is , and each angle is .
Since the sum of the exterior angles is , we just need to subtract the given angle values to find the unknown angle:
The unknown angle is .

EXAMPLE 2

Nice work! Now let's try an irregular shape, where the sides and angles are not equal. Find the missing angle in this shape:

Step 1. Determine whether you’re looking for interior or exterior angles.

If we think of the shape as a house, the interior angles live inside of the house, while the exterior angles live in exile outside of the house.
We're looking for the exterior angles in this problem.

Step 2. Calculate the sum of exterior angles.

The sum of exterior angles for all polygons is always .This animation ⬆️ can help you see how the exterior angles combine to create a circle, which is .

Step 3. Find the value of an individual angle.

Since the sum of the exterior angles is , we just need to subtract the given angle values to find the unknown angle:
The unknown angle is .
Awesome! If you’d like to practice additional problems with exterior angles, expand the
Practice
below before closing out this lesson! ⚡
PRACTICE
Exterior Angles

Practice: Exterior Angles

Question 1 of 6:Find the value of an exterior angle in a regular 6-sided shape.

Step 1. Determine whether you’re looking for an interior or exterior angle.

Are we looking for an interior or exterior angle?

Step 2. Calculate the sum of exterior angles.

The sum of exterior angles for all polygons is always .

Step 3. Find the value of the individual angle.

Since we're told the shape is regular, which formula can we use to determine the value of each angle?
Nice! Now let's plug in:

Great work! The value of the missing angle is   !
CONCLUSION
Dang, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?
Leave Feedback