A polygon is simply a plain figured enclosed by straight lines. In Greek, **poly **means *many *and **gon **means *angle*. The easiest polygon is a triangle which has actually 3 sides and also 3 angle which sum up to 180 degrees. Here, the diagonal line of a polygon formula is offered with description and also solved examples.

There have the right to be countless sided polygons and also they have the right to either be regular (equal length and interior angles) or irregular. A polygon have the right to be more classified as concave or convex based on its interior angles. If the internal angles are much less than 180 degrees, the polygon is ** convex**, otherwise, that is a

**polygon. It need to be detailed the sides of a polygon are constantly a directly line.**

*concave*You are watching: How many diagonals does a polygon have

In a polygon, the diagonal line is the heat segment the joins 2 non-adjacent vertices. An interesting fact about the diagonals of a polygon is that in concave polygons, at least one diagonal is actually exterior the polygon.

Now, for an “n” sided-polygon, the variety of diagonals deserve to be acquired by the complying with formula:

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

This formula is simply formed by the mix of diagonals that each vertex sends to another vertex and also then individually the total sides. In various other words, one n-sided polygon has n-vertices which can be joined through each various other in nC2 ways.

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

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

**Example 1:**

*Find the total number of diagonals included in an 11-sided continuous polygon.*

**Solution:**

In one 11-sided polygon, total vertices space 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 sides also. So, individually the sides will give the total diagonals contained by the polygon.

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

**Formula Method:**

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

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

**Example 2:**

*In a 20-sided polygon, one **vertex does no send any diagonals. Discover out how countless diagonals does the 20-sided polygon contain.*

**Solution:**

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

But, since one peak does not send any type of diagonals, the diagonals by the vertex requirements to it is in subtracted from the total number of diagonals.

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In a polygon, it is known that each vertex provides (n-3) diagonals. In this polygon, each vertex provides (20-3) = 17 diagonals.

Now, since 1 peak does no send any type of diagonal, the complete diagonal in this polygon will certainly be (170-17) = 153 diagonals.