How To Calculate The Sum Of Interior Angles: 8 Steps - WikiHow

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Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow How to Calculate the Sum of Interior Angles PDF download Download Article Co-authored by David Jia

Last Updated: March 10, 2025 References

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  • Using the Formula
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  • Drawing Triangles
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  • Video
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  • Q&A
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  • Tips
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  • Things You'll Need
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This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 439,674 times.

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.

Steps

Method 1 Method 1 of 2:

Using the Formula

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  1. Step 1 Set up the formula for finding the sum of the interior angles. 1 Set up the formula for finding the sum of the interior angles. The formula is s u m = ( n − 2 ) × 180 {\displaystyle sum=(n-2)\times 180} , where s u m {\displaystyle sum} is the sum of the interior angles of the polygon, and n {\displaystyle n} equals the number of sides in the polygon.[1] [2]
    • The value 180 comes from how many degrees are in a triangle. The other part of the formula, n − 2 {\displaystyle 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]
    • 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] 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. Step 2 Count the number of sides in your polygon. 2 Count the number of sides in your polygon. Remember that a polygon must have at least three straight sides.[5]
    • 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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  3. Step 3 Plug the value of n {\displaystyle n} into the formula. 3 Plug the value of n {\displaystyle n} into the formula. Remember, n {\displaystyle n} is the number of sides in your polygon.[6]
    • For example, if you are working with a hexagon, n = 6 {\displaystyle n=6} , since a hexagon has 6 sides. So, your formula should look like this: s u m = ( 6 − 2 ) × 180 {\displaystyle sum=(6-2)\times 180}
  4. Step 4 Solve for n {\displaystyle n} . 4 Solve for n {\displaystyle 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]
    • For example, to find out the sum of the interior angles of a hexagon, you would calculate: s u m = ( 6 − 2 ) × 180 {\displaystyle sum=(6-2)\times 180} s u m = ( 4 ) × 180 {\displaystyle sum=(4)\times 180} s u m = ( 4 ) × 180 = 720 {\displaystyle sum=(4)\times 180=720} So, the sum of the interior angles of a hexagon is 720 degrees.
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Method 2 Method 2 of 2:

Drawing Triangles

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  1. Step 1 Draw the polygon whose angles you need to sum. 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. Step 2 Choose one vertex. 2 Choose one vertex. Label this vertex A.
    • A vertex is a point where two sides of a polygon meet.
  3. Step 3 Draw a straight line from Point A to each other vertex in the polygon. 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. Step 4 Multiply the number of triangles you created by 180. 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]
    • For example, since you divided your hexagon into 4 triangles, you would calculate 4 × 180 = 720 {\displaystyle 4\times 180=720} to find a total of 720 degrees in the interior of your polygon.
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Community Q&A

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  • Question How do I find a single interior angle? Community Answer Community Answer Work out what all the interior adds up to, then divide by however many sides the shape has. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 35 Helpful 46
  • Question How do I calculate the number of sides of a polygon if the sum of the interior angles is 1080? Donagan Donagan Top Answerer Divide that sum by 180°, then add 2. In this example, 1080° / 180° = 6. 6 + 2 = 8. The polygon has 8 sides. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 19 Helpful 40
  • Question 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 Donagan Top Answerer 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°. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 20 Helpful 20
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Video

Tips

  • 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. Thanks Helpful 1 Not Helpful 3
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Things You'll Need

  • Pencil
  • Paper
  • Protractor (optional)
  • Pen
  • Eraser
  • Ruler

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References

  1. ↑ David Jia. Academic Tutor. Expert Interview
  2. ↑ http://www.mathsisfun.com/geometry/interior-angles-polygons.html
  3. ↑ https://www.bbc.co.uk/bitesize/guides/zshb97h/revision/6
  4. ↑ https://www.bbc.co.uk/bitesize/guides/zshb97h/revision/6
  5. ↑ https://www.bbc.co.uk/bitesize/guides/zshb97h/revision/6
  6. ↑ https://www.mathsisfun.com/geometry/interior-angles-polygons.html
  7. ↑ https://www.mathsisfun.com/geometry/interior-angles-polygons.html
  8. ↑ https://www.cuemath.com/geometry/sum-of-angles-in-a-polygon/

About This Article

David Jia Co-authored by: David Jia Math Tutor This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 439,674 times. 16 votes - 76% Co-authors: 17 Updated: March 10, 2025 Views: 439,674 Categories: Geometry Article SummaryX

To calculate the sum of interior angles, start by counting the number of sides in your polygon. Next, plug this number into the formula for the "n" value. Then, solve for "n" by subtracting 2 from the number of sides and multiplying the difference by 180. This will give you, in degrees, the sum of the interior angles in your polygon! To learn how to calculate the sum of interior angles by drawing triangles, read on! Did this summary help you?YesNo

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Reader Success Stories

  • Hannah Mark

    Hannah Mark

    Nov 29, 2017

    "When you explained slowly how the formula works, I completely understood. I have a big math test tomorrow, and I..." more
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Did this article help you?

Yes No Advertisement Cookies make wikiHow better. By continuing to use our site, you agree to our cookie policy. David Jia Co-authored by: David Jia Math Tutor 16 votes - 76% Click a star to vote Co-authors: 17 Updated: March 10, 2025 Views: 439,674 Hannah Mark

Hannah Mark

Nov 29, 2017

"When you explained slowly how the formula works, I completely understood. I have a big math test tomorrow, and I..." more Holly Flavian

Holly Flavian

May 2, 2020

"This has really helped me in remembering what the teacher taught. Thanks a lot." Afia Subah

Afia Subah

May 18, 2017

"The formula is super helpful, thank you." Share yours! More success stories Hide success stories

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