How to Calculate the Sum of Interior Angles
A polygon is any closed figure with sides made from straight lines. At each vertex of a polygon,
there is both an interior and exterior angle, corresponding to the angles on the inside and outside
of the closed figure. Understanding the relationships that govern these angles is useful in various
geometrical problems. In particular, it is helpful to know how to calculate the sum of interior
angles in a polygon. This can be done using a simple formula.
Exterior angle
Interior angle
Hexagon = 6 sides
1. Count the number of sides your polygon has. The method for calculating the sum of
interior angles is based on how many sides the polygon has. Remember that a polygon
must have at least 3 sides (a triangle), and each side must be a straight line.
Hexagon = 6 sides
6–2=4
2. Subtract 2 from the number of sides. For example, subtracting 2 from a triangle gives
you the number 1. Subtracting 2 from a pentagon (which has 5 sides) gives you the
number 3. Subtracting 2 from a hexagon (which has 6 sides) gives you the number 4.
4
Hexagon = 6 sides
6–2=4
3
1
2
4 x 180º = 720º
Notice that there are 4 triangles and remember that the sum of each triangle measures 180º
This is why this formula works!
3. Multiply this number by 180. Multiply the number arrived at in the previous step (4) by
180. This will give you the sum of the polygon's interior angles, expressed in degrees.
For example, consider this hexagon. Subtracting 2 from a hexagon's 6 sides equals 4.
Multiplying 4 by 180 equals 720. Therefore, a hexagon (regular or irregular-that means
the polygons don’t have to have equal sides or angles for this to work) has interior angles
that add up to 720 degrees. Because this hexagon IS a regular polygon, you could then
divide by the number of interior angles to find out what each angle measures.
4. Review the formula used to calculate this sum. Building a formula from the steps
above yields: s = 180(n - 2), where "s" is the sum of the interior angles and "n" is the
polygon's number of sides. This formula can be used for a polygon with any number of
sides.
1
Hexagon = 6 sides
2
6
3
6–2=4
4 x 180º = 720º
5
4
720º ÷ 6 = 120º
Each angle in this regular polygon (a hexagon) = 120º

It does not matter whether the polygon is regular or irregular, nor does it matter what the
individual interior angles measure. Given a certain number of sides, a polygon's interior
angles can always be summed using the formula above.
5. Derive the formula for further understanding. If you forget the formula above, or
simply want to understand why it works, you can derive it fairly easily. This is done by
segmenting any polygon into triangles.

Remember that a triangle's 3 interior angles will add up to 180 degrees, no matter what
the shape of the triangle. This is the premise upon which the formula above can be built.

Draw a polygon on a sheet of paper. For example, consider a square, which is a 4-sided polygon.
You can split the square up into 2 triangles by drawing a line connecting 2 of its opposite
corners. Note that the interior angles of each triangle can be added to equal the interior angles
of the original polygon.

Once you have split your polygon up into separate triangles, count the number of triangles in
the polygon. Multiplying this number by 180 will yield the sum of the polygon's interior angles.
The formula presented above makes use of this method.