Content

• To step
• Method 1 of 2: Draw the diagonals
• Method 2 of 2: Using the formula for the diagonal

Finding diagonals in a polygon is a necessary skill to advance in mathematics. It may seem difficult at first, but is quite easy once you learn the basic formula. A diagonal is any segment drawn between vertices of a polygon that does not contain the sides of that polygon. A polygon is any shape that has more than three sides. Using a very simple formula, you can calculate the number of diagonals in each polygon, whether it has four sides or 4000 sides.

To step

Method 1 of 2: Draw the diagonals

1. Know the names of the different polygons. You may first need to determine how many sides the polygon has. Each polygon has a prefix that indicates the number of sides. Here are the names of polygons up to twenty sides:
   • Four-sided / tetragonic: 4 sides
   • Pentagon / pentagon: 5 sides
   • Hexagon / hexagon: 6 sides
   • Heptagon: 7 sides
   • Octagon / octagon: 8 sides
   • Nonagon / Enneagon: 9 sides
   • Decagon: 10 sides
   • Hendecagon: 11 sides
   • Dodecagon: 12 sides
   • Triskaidecagoon: 13 sides
   • Tetradecagon: 14 sides
   • Pentadecagon: 15 sides
   • Hexadecagon: 16 sides
   • Heptadecagon: 17 sides
   • Octadecagon: 18 sides
   • Ennea Decagon: 19 sides
   • Icosagoon: 20 sides
   • Note that a triangle has no diagonals.
2. Draw the polygon. If you want to know how many diagonals there are in a square, start by drawing the square. The easiest way to find and count diagonals is to draw the polygon symmetrically, with each side having the same length. It is important to note that even if the polygon is not symmetrical, it still has the same number of diagonals.
   • To draw the polygon, use a ruler and draw each side the same length, connecting all sides.
   • If you are not sure what the polygon looks like, search for images online. For example, a stop sign is an octagon.
3. Draw the diagonals. A diagonal is a segment that is drawn from one corner of the shape to the other, except for the sides of the polygon. Use a ruler to draw a diagonal to any other available vertex.
   • For a square, draw a 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 for easier counting.
   • Note that this method becomes much more difficult with polygons with more than ten sides.
4. Count the diagonals. There are two options for counting diagonals: you can count them when you draw the diagonals or when they are drawn. When counting each diagonal, write a small number above the diagonal to indicate that it has been counted. It is easy to lose track while counting if there are many diagonals mixed up.
   • For the square, there are two diagonals: one diagonal for every two vertices.
   • A hexagon has nine diagonals: there are three diagonals for every three vertices.
   • A heptagon has 14 diagonals. Beyond the heptagon, it becomes more difficult to count the diagonals because there are so many diagonals.
5. Be careful not to count diagonals more than once. Each vertex can have multiple diagonals, but that does not mean that the number of diagonals is equal to the number of vertices times the number of diagonals. When counting the diagonals, make sure you only count each diagonal once.
   • For example, a pentagon (five sides) has only five diagonals. Each vertex has two diagonals, so if you counted every diagonal of each vertex twice, you would think there are 10 diagonals. This is incorrect because you have counted each diagonal twice!
6. Practice with some examples. Draw some other polygons and count the number of diagonals. The polygon does not have to be symmetrical for this method to work.In the case of a hollow polygon, you may need to draw some diagonals outside the actual polygon.
   • A hexagon or hexagon has 9 diagonals.
   • A heptagon has 14 diagonals.

Method 2 of 2: Using the formula for the diagonal

1. Define 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. Using the distributive property, this can be rewritten as (n - 3n) / 2. You can look at it in both directions, both equations are identical.
   • 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 has no diagonals.
2. Determine the number of sides of the polygon. To use this formula, you need to know the number of sides of the polygon. The number of sides is given in the name of the polygon, so you just need to know what each name means. Here are some common prefixes you may encounter with polygons:
   • 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 polygons with many sides, you can just see "n-goon", where "n" is the number of sides. For example, a 44-sided polygon is written as 44-goon.
   • If you get a picture of the polygon, you can simply count the number of sides.
3. Include the number of sides in the equation. Once you know how many sides the polygon has, all you have to do is put that number in the equation and solve the equation. Wherever you see "n" in the equation, the number of sides of the polygon is replaced by the number of sides of the polygon.
   • For example: A dodecagon has 12 sides.
   • Write the equation: n (n-3) / 2
   • Process this in the variable: (12 (12 - 3)) / 2
4. Solve the equation. Finally, solve the equation in the correct order of operations. Start by solving the subtraction, then the multiplication, and finally the division. The last answer is the number of diagonals the polygon has.
   • For example: (12 (12 - 3)) / 2
   • Subtract: (12 * 9) / 2
   • Multiply: (108) / 2
   • Share: 54
   • So a dodecagon has 54 diagonals.
5. Practice with more examples. The more practice you have with a math concept, the better you can use it. Working out many practice items 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 with any number of sides greater than three.
   • 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-goon (96 sides): 96 (96-3) / 2 = 96 * 93/2 = 8928/2 = 4464 diagonals.