And now we are back to polygons. There are two variations of questions that come up concerning the number of diagonals in a polygon. First, you may be asked how many diagonals can be drawn from a single vertex of an n-gon. More commonly though, you will be asked how many distinct diagonals (total) can be drawn inside an n-gon.
This concept appeared 11 times last year, with a median placement at question # 51.
Number Dojo Level: 188
To succeed with this concept, you will need to know how many sides are in each type of polygon, which we covered in AngleExterior. And you will need to use a little bit of logic. I will teach you how these questions work and why, and then I will give you a formula to memorize if you choose.
From a Single Vertex:
Think about a triangle...have you ever seen a diagonal drawn inside a triangle? The answer should be no, because if you start at one vertex and try to draw a line to another vertex, you will stay on the perimeter of the triangle. (In other words, the only 3 vertices of a triangle are the two adjacent ones and the original one, which would keep you on the edge of the triangle).
If you start at a specific vertex in a square, the only diagonal you can draw is to the opposite corner (forming two 45-45-90 right triangles). Otherwise you are drawing a line on top of the sides of the square.
So to determine how many diagonals can be drawn from a single vertex of an n-gon, where n is the number of sides of the polygon, simply subtract 3 from n. So: n - 3.
From Any Vertex:
Continuing with this logic, you might think that you can multiply (n - 3) by n to determine the total number of diagonals in a polygon. But remember that each diagonal touches 2 vertices (so every vertex shares each diagonal with another vertex). So we have to divide by 2 to avoid duplication.
Imagine that you label the 5 vertices of a pentagon A - E, going clockwise. Starting at vertex A, you have the diagonals AC and AD. From vertex B, you have the diagonals BD and BE. From vertex C, you have the diagonals CE and CA. And so on. But CA is the same as AC, so we divide by 2 to determine the number of distinct diagonals. Put another way, here are the distinct diagonals of the pentagon:
- AC (which is the same as CA)
- AD (which is the same as DA)
- BD (which is the same as DB)
- BE (which is the same as EB)
- CE (which is the same as EC).
Example 1: How many diagonals can be drawn from one vertex of an octagon?
- An octagon has 8 sides. Subtract 3 to get 5.
Example 2: How many diagonals can be drawn from a single vertex of an icosagon?
- An icosagon has 20 sides. 20 - 3 = 17.
Example 3: How many diagonals can be drawn from one vertex of a dodecagon?
- A dodecagon has 12 sides. 12 - 3 = 9.
Example 4: How many distinct diagonals can be drawn inside a 15-sided polygon?
- Use the formula n(n - 3)/2. We are given n = 15.
- 15(15 - 3)/2 = 15(12)/2 = 15 x 6 = 90.
Example 5: How many distinct diagonals can be drawn inside a heptagon?
- A heptagon has 7 sides.
- 7(7 - 3)/2 = 7(4)/2 = 7 x 2 = 14.
Example 6: The number of distinct diagonals which can be drawn inside an undecagon is ___
- An undecagon has 11 sides.
- 11(11 - 3)/2 = 11(8)/2 = 11 x 4 = 44.