How to Calculate the Sum of Interior Angles

Co-authored by David Jia

Last Updated: October 3, 2022 References

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, or by dividing the polygon into triangles.

Method 1

Method 1 of 2:

Using the Formula

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1
Set up the formula for finding the sum of the interior angles. The formula is $sum=(n-2)\times 180$ , where $sum$ is the sum of the interior angles of the polygon, and $n$ equals the number of sides in the polygon. ^{[1]
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David Jia
Academic Tutor

Expert Interview. 23 February 2021} ^{[2]
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- The value 180 comes from how many degrees are in a triangle. The other part of the formula, $n-2$ is a way to determine how many triangles the polygon can be divided into. So, essentially the formula is calculating the degrees inside the triangles that make up the polygon. ^{[3]
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- This method will work whether you are working with a regular or irregular polygon. Regular and irregular polygons with the same number of sides will always have the same sum of interior angles, the difference only being that in a regular polygon, all interior angles have the same measurement. ^{[4]
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  Research source} In an irregular polygon, some of the angles will be smaller, some of the angles will be larger, but they will still add up to the same number of degrees that are in the regular shape.
2
Count the number of sides in your polygon. Remember that a polygon must have at least three straight sides. ^{[5]
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- For example, if you want to know the sum of the interior angles of a hexagon, you would count 6 sides.
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Plug the value of $n$ into the formula. Remember, $n$ is the number of sides in your polygon. ^{[6]
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- For example, if you are working with a hexagon, $n=6$ , since a hexagon has 6 sides. So, your formula should look like this:
  $sum=(6-2)\times 180$
4
Solve for $n$ . To do this, subtract 2 from the number of sides, and multiply the difference by 180. This will give you, in degrees, the sum of the interior angles in your polygon. ^{[7]
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- For example, to find out the sum of the interior angles of a hexagon, you would calculate:
  $sum=(6-2)\times 180$
  $sum=(4)\times 180$
  $sum=(4)\times 180=720$
  So, the sum of the interior angles of a hexagon is 720 degrees.

Method 2

Method 2 of 2:

Drawing Triangles

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1
Draw the polygon whose angles you need to sum. The polygon can have any number of sides and can be regular or irregular.
- For example, you might want to find the sum of the interior angles of a hexagon, so you would draw a six-sided shape.
2
Choose one vertex. Label this vertex A.
- A vertex is a point where two sides of a polygon meet.
3
Draw a straight line from Point A to each other vertex in the polygon. The lines should not cross. You should create a number of triangles.
- You do not have to draw lines to the adjacent vertices, since they are already connected by a side.
- For example, for a hexagon you should draw three lines, dividing the shape into 4 triangles.
4
Multiply the number of triangles you created by 180. Since there are 180 degrees in a triangle, by multiplying the number of triangles in your polygon by 180, you can find the sum of the interior angles of your polygon. ^{[8]
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- For example, since you divided your hexagon into 4 triangles, you would calculate $4\times 180=720$ to find a total of 720 degrees in the interior of your polygon.

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Question

How do I find a single interior angle?

Community Answer

Work out what all the interior adds up to, then divide by however many sides the shape has.

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How do I calculate the number of sides of a polygon if the sum of the interior angles is 1080?

Donagan

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Divide that sum by 180°, then add 2. In this example, 1080° / 180° = 6. 6 + 2 = 8. The polygon has 8 sides.

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If two equilateral triangles are placed together to form a rhombus, how do I calculate the value of each interior angle of this rhombus, and how do I find the sum?

Donagan

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In the rhombus you describe, the two smaller interior angles would each be 60°, and the two larger interior angles would each be 120°. You wouldn't have to calculate the angles. Simple inspection of the rhombus and the two triangles would show what the angles are, given that equilateral triangles have three 60° angles. The sum is 60° + 60° + 120° + 120°.

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Check your work on a piece of paper using a protractor to sum the interior angles manually. When doing this, be careful while drawing the polygon's sides as they should be linear.

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How to Calculate the Sum of Interior Angles

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Drawing Triangles

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