How to Find How Many Diagonals Are in a Polygon: 11 Steps

This article was co-authored by Jake Adams. Jake Adams is an academic tutor and the owner of Simplifi EDU, a Santa Monica, California based online tutoring business offering learning resources and online tutors for academic subjects K-College, SAT & ACT prep, and college admissions applications. With over 14 years of professional tutoring experience, Jake is dedicated to providing his clients the very best online tutoring experience and access to a network of excellent undergraduate and graduate-level tutors from top colleges all over the nation. Jake holds a BS in International Business and Marketing from Pepperdine University.

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Finding diagonals in a polygon is a necessary skill to develop in math. It may seem difficult at first, but is pretty simple once you learn the basic formula. A diagonal is any line segment drawn between vertices of a polygon that doesn’t include the sides of that polygon.[1] X Research source A polygon is any shape that has more than three sides. Using a very simple formula, you can calculate the number of diagonals in any polygon, whether it has 4 sides or 4,000 sides.

Steps

Drawing the Diagonals

1. 1Know the names of polygons. You may need to first identify how many sides are present in the polygon. Each polygon has a prefix that indicates the number of sides it has. Here are the names of polygons with up to twenty sides:[2] X Research source
   • Quadrilateral/tetragon: 4 sides
   • Pentagon: 5 sides
   • Hexagon: 6 sides
   • Heptagon: 7 sides
   • Octagon: 8 sides
   • Nonagon/Enneagon: 9 sides
   • Decagon: 10 sides
   • Hendecagon: 11 sides
   • Dodecagon: 12 sides
   • Triskaidecagon/tridecagon: 13 sides
   • Tetrakaidecagon/tetradecagon: 14 sides
   • Pentadecagon: 15 sides
   • Hexadecagon: 16 sides
   • Heptadecagon: 17 sides
   • Octadecagon: 18 sides
   • Enneadecagon: 19 sides
   • Icosagon: 20 sides
   • Note that a triangle has no diagonals.[3] X Research source
2. 2Draw the polygon. If you wanted to know how many diagonals were present in a square, you would start by drawing the square. The easiest way to find diagonals and count them is to draw the polygon symmetrically, each side has the same length. It’s important to note that even if the polygon is not symmetrical, it will still have the same number of diagonals.[4] X Research source
   • To draw the polygon, use a ruler and draw each side the same length, connecting all of the sides together.
   • If you’re unsure what the polygon will look like, search for pictures online. For example, a stop sign is an octagon.

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3. 3Draw the diagonals. A diagonal is a line segment drawn from one corner of the shape to another, excluding the sides of the polygon.[5] X Research source Starting at one vertex of the polygon, use a ruler to draw a diagonal to every other available vertex.
   • For a square, draw one line from the bottom left corner to the top right corner and another line from the bottom right corner to the top left corner.
   • Draw diagonals in different colors to make them easier to count.
   • Note that this method gets much more difficult with polygons that have more than ten sides.
4. 4Count the diagonals. There are two options for counting: you can count as you draw the diagonals or count them once they have been drawn. As you count each diagonal, draw a small number above the diagonal to denote that it has been counted. It is easy to lose track while counting when there are a lot of diagonals crossing each other.
   • For the square, there are two diagonals: one diagonal for every two vertices.
   • A hexagon has 9 diagonals: there are three diagonals for every three vertices.
   • An octagon has 20 diagonals. Past the heptagon, it gets more difficult to count the diagonals because there are so many of them.
5. 5Beware of counting a diagonal more than once. Each vertex may have multiple diagonals, but that doesn’t mean that the number of diagonals is equal to the number of vertices times the number of diagonals. Take care when counting the diagonals to count each one only once.[6] X Research source
   • For example, a pentagon (5 sides) has only 5 diagonals. Each vertex has two diagonals, so if you counted each diagonal from every vertex twice, you might think there were 10 diagonals. This is incorrect because you would have counted each diagonal twice!
6. 6Practice with some examples. Draw some other polygons and count the number of diagonals. The polygon does not have to be symmetric for this method to work. In the case of a concave polygon, you may have to draw some of the diagonals outside the actual polygon.[7] X Research source
   • A hexagon has 9 diagonals.
   • A octagon has 20 diagonals.

Using the Diagonal Formula

1. 1Define the formula. The formula to find the number of diagonals of a polygon is n(n-3)/2 where “n” equals the number of sides of the polygon.[8] X Expert SourceJake AdamsAcademic Tutor & Test Prep Specialist Expert Interview. 20 May 2020. Using the distributive property this can be rewritten as (n2 – 3n)/2. You may see it either way, both equations are identical.[9] X Research source
   • This equation can be used to find the number of diagonals of any polygon.
   • Note that the triangle is an exception to this rule. Due to the shape of the triangle, it does not have any diagonals.[10] X Research source
2. 2Identify the number of sides in the polygon. To use this formula, you must identify the number of sides that the polygon has. The number of sides is given in the name of the polygon, you just need to know what each name means. Here are some of common prefixes you will see in polygons:[11] X Research source
   • Tetra (4), penta (5), hexa (6), hepta (7), octa (8), ennea (9), deca (10), hendeca (11), dodeca (12), trideca (13), tetradeca (14), pentadeca (15), etc.
   • For very large sided polygons you may simply see it written “n-gon”, where “n” is the number of sides. For example, a 44-sided polygon would be written as 44-gon.
   • If you are given a picture of the polygon, you can simply count the number of sides.
3. 3Plug the number of sides into the equation.[12] X Expert SourceJake AdamsAcademic Tutor & Test Prep Specialist Expert Interview. 20 May 2020. Once you know how many sides the polygon has, you just need to plug that number into the equation and solve. Everywhere you see “n” in the equation will be replaced with the number of sides of the polygon.[13] X Research source
   • For example: A dodecagon has 12 sides.
   • Write the equation: n(n-3)/2
   • Plug in the variable: (12(12 – 3))/2
4. 4Solve the equation. Finish by solving the equation using the proper order of operations. Start by solving the subtraction, then multiply, then divide. The final answer is the number of diagonals the polygon has.[14] X Research source
   • For example: (12(12 – 3))/2
   • Subtract: (12*9)/2
   • Multiply: (108)/2
   • Divide: 54
   • A dodecagon has 54 diagonals.
5. 5Practice with more examples. The more practice you have with a math concept, the better you will be at using it. Doing lots of examples will also help you memorize the formula in case you need it for a quiz, test, or exam. Remember, this formula works for a polygon of any number of sides greater than 3.[15] X Research source
   • Hexagon (6 sides): n(n-3)/2 = 6(6-3)/2 = 6*3/2 = 18/2 = 9 diagonals.
   • Decagon (10 sides): n(n-3)/2 = 10(10-3)/2 = 10*7/2 = 70/2 = 35 diagonals.
   • Icosagon (20 sides): n(n-3)/2 = 20(20-3)/2 = 20*17/2 = 340/2 = 170 diagonals.
   • 96-gon (96 sides): 96(96-3)/2 = 96*93/2 = 8928/2 = 4464 diagonals.

Expert Q&A
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• QuestionWhat is the formula to find the number of diagonals?Jake AdamsJake Adams is an academic tutor and the owner of Simplifi EDU, a Santa Monica, California based online tutoring business offering learning resources and online tutors for academic subjects K-College, SAT & ACT prep, and college admissions applications. With over 14 years of professional tutoring experience, Jake is dedicated to providing his clients the very best online tutoring experience and access to a network of excellent undergraduate and graduate-level tutors from top colleges all over the nation. Jake holds a BS in International Business and Marketing from Pepperdine University.
  Academic Tutor & Test Prep SpecialistAcademic Tutor & Test Prep SpecialistExpert Answer
• QuestionHow many diagonals can be drawn from one vertex of nonagon?Community AnswerYou can draw six, one for each of the vertices, except for the vertex you’re drawing from, and the two adjacent vertices.
• QuestionWhat is the relationship between the number of sides in an icosikaipentagon and the number of diagonals?DonaganTop AnswererIt is the same relationship for any polygon, as expressed in the formula n(n-3)/2.

References

1. ↑ https://www.mathsisfun.com/geometry/polygons-diagonals.html
2. ↑ https://www.mathsisfun.com/geometry/polygons.html
3. ↑ https://www.cuemath.com/geometry/diagonals/
4. ↑ https://www.mathsisfun.com/geometry/polygons-diagonals.html
5. ↑ https://www.mathsisfun.com/geometry/polygons-diagonals.html
6. ↑ http://www.mathopenref.com/polygondiagonal.html
7. ↑ https://www.mathsisfun.com/geometry/polygons-diagonals.html
8. ↑ Jake Adams. Academic Tutor & Test Prep Specialist. Expert Interview. 20 May 2020.
9. ↑ https://www.cuemath.com/diagonal-of-a-polygon-formula/

1. ↑ https://www.cuemath.com/geometry/diagonals/
2. ↑ http://www.infoplease.com/ipa/A0881983.html
3. ↑ Jake Adams. Academic Tutor & Test Prep Specialist. Expert Interview. 20 May 2020.
4. ↑ http://www.mathopenref.com/polygondiagonal.html
5. ↑ http://www.mathopenref.com/polygondiagonal.html
6. ↑ https://www.brightstorm.com/math/geometry/reasoning-diagonals-angles-and-parallel-lines/number-of-diagonals-in-a-polygon/

About This Article

To find out how many diagonals a polygon has, first count the number of sides, or straight lines, that make up the polygon. Then, subtract 3 from the number of sides. Next, multiply that number by the number of sides. Finally, divide the answer by 2, and you’ll have the number of diagonals within the polygon. For example, if a polygon has 6 sides, you’d find it has 9 diagonals. For an alternate way to determine the number of diagonals in a polygon, read on!

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