A polygon is just a level figured attached by right lines. In Greek, poly means many and gon means angle. The easiest polygon is a triangle which has actually 3 sides and 3 angles which sum up to 180 degrees. Here, the diagonal of a polygon formula is offered with description and solved examples.


There have the right to be numerous sided polygons and also they can either be constant (equal length and interior angles) or irregular. A polygon can be additional classified as concave or convex based upon its interior angles. If the internal angles are less than 180 degrees, the polygon is convex, otherwise, the is a concave polygon. It need to be listed the sides of a polygon are always a directly line.

You are watching: Total number of diagonals in a polygon

In a polygon, the diagonal line is the line segment that joins two non-adjacent vertices. An amazing fact around the diagonals the a polygon is that in concave polygons, at the very least one diagonal line is actually external the polygon.


Now, for an “n” sided-polygon, the number of diagonals have the right to be acquired by the following formula:

Number of Diagonals = n(n-3)/2

This formula is simply formed by the mix of diagonals that each vertex sends to one more vertex and then subtracting the full sides. In various other words, one n-sided polygon has n-vertices which have the right to be joined through each other in nC2 ways.

Now by individually n v nC2 ways, the formula obtained is n(n-3)/2.

For example, in a hexagon, the complete sides room 6. So, the total diagonals will be 6(6-3)/2 = 9.


Example 1:

Find the total number of diagonals included in one 11-sided constant polygon.


In one 11-sided polygon, complete vertices room 11. Now, the 11 vertices have the right to be joined through each other by 11C2 methods i.e. 55 ways.

Now, there space 55 diagonals feasible for an 11-sided polygon which includes its political parties also. So, subtracting the sides will give the complete diagonals had by the polygon.

So, complete diagonals consisted of within one 11-sided polygon = 55 -11 i.e. 44.

Formula Method:

According to the formula, variety of diagonals = n (n-3)/ 2.

So, 11-sided polygon will certainly contain 11(11-3)/2 = 44 diagonals.

Example 2:

In a 20-sided polygon, one vertex does no send any kind of diagonals. Uncover out how numerous diagonals does that 20-sided polygon contain.


In a 20-sided polygon, the total diagonals are = 20(20-3)/2 = 170.

But, due to the fact that one peak does not send any type of diagonals, the diagonals by that vertex needs to be subtracted native the total variety of diagonals.

See more: Which Months Have 5 Sundays In 2018, How Many Sundays In 2018

In a polygon, that is recognized that each vertex renders (n-3) diagonals. In this polygon, every vertex renders (20-3) = 17 diagonals.

Now, because 1 vertex does not send any diagonal, the total diagonal in this polygon will be (170-17) = 153 diagonals.