Diagonals of Polygon Formulas

Formula to Find Diagonal of a Polygon

In Geometry, diagonals of a polygon are line segments joining two non-adjacent vertices.

Example:

A square given below has two pairs of non-adjacent vertices.

By joining the vertices of these two pairs, we get two diagonals of a square.


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In this square ABCD, A and C, and B and D are two pairs of adjacent vertices.

Hence, AC and BD are diagonals of a square.

The formula to find diagonal of a polygon square is:

\[\frac{n(n - 3)}{2}\], where n is the number of sides.

We can use this formula to find the diagonals of a polygon with any number of sides.

In this article, we will discuss the diagonals of a polygon formula and the formula to find the number of diagonals in a polygon.


What is a Polygon?

A simple closed figure made up of line segments is known as a polygon. Polygons are generally classified according to the number of vertices and sides they have. For example, a triangle is a polygon with 3 sides and 3 vertices, similarly, a quadrilateral is a polygon with 4 vertices and 4 sides.


Formulas for Number of Diagonals in a Polygon

Can you think of the number of diagonals of a quadrilateral, heptagon, hexagon, octagon, nonagon, etc?

You can find the number of diagonals of different polygons using the number of diagonals formula.

As you can see, it is quite difficult to draw diagonals of a polygon with a large number of sides.

To easily find the number of diagonals of a polygon with a large number of sides, we can use the formulas to find the number of diagonals in a polygon with n sides without actually drawing them.


The formula to find diagonals of a polygon with n side is:


\[\frac{n(n - 3)}{2}\]

Where n represents the total number of sides of the polygon.


The following table shows the number of diagonals of different polygons which is calculated using the formula of diagonals of a polygon.


Number of Diagonals of Polygons Table

Geometric Shapes

Number of Sides (n)

Diagonal of Polygon Formula

Triangle

3

\[\frac{3(3 - 3)}{2} = 0\]

Quadrilateral

4

\[\frac{4(4 - 3)}{2} = 2\]

Heptagon

5

\[\frac{5(5 - 3)}{2} = 5\]

Hexagon

6

\[\frac{6(6 - 3)}{2} = 9\]

Hectagon

7

\[\frac{7(7 - 3)}{2} = 14\]

Octagon

8

\[\frac{8(8 - 3)}{2} = 20\]

Nonagon

9

\[\frac{9(9 - 3)}{2} = 27\]

Decagon

10

\[\frac{10(10 - 3)}{2} = 35\]

Hendecagon

11

\[\frac{11(11 - 3)}{2} = 44\]

Dodecagon

12

\[\frac{12(12 - 3)}{2} = 54\]

Tridecagon

13

\[\frac{13(13 - 3)}{2} = 65\]

Tetradecagon

14

\[\frac{14(14 - 3)}{2} = 77\]

Pentadecagon

15

\[\frac{15(15 - 3)}{2} = 90\]


Solved Examples Using Formula for Number of Diagonals in a Polygon

1.  Find the number of diagonals of a hexadecagon.

Solution:

The number of sides of an hexadecagon, is n = 16

The number of diagonals of octagon can be found by using the no. of diagonals formula \[\frac{n(n - 3)}{2}\].

As we know, the total number of sides of a hexadecagon is 16. Substituting the values in the formula of diagonal of polygon we get,

\[\frac{n(n - 3)}{2}\]

\[= \frac{16(16 - 3)}{2}\]

\[= \frac{16(13)}{2}\]

\[= \frac{208}{2}\]

= 104

The number of diagonals of an octagon = 104.


2. Find the number of sides of a polygon with 90 diagonals?

Solution:

Let the number of sides of a polygon with 90 diagonals = x

Total number of diagonals = 90

The number of sides of of a polygon with 90 diagonals can be found by using the number of diagonals formula \[\frac{n(n - 3)}{2}\].

As we know, the total number of sides of a hexadecagon is x. Substituting the values in the formula of diagonal of polygon we get,

\[\frac{n(n - 3)}{2}\]

\[\frac{x(x - 3)}{2} = 90\]

\[\frac{x^{2} - 3x}{2} = 90\]

\[x^{2} - 3x = 180\]

\[x^{2} - 3x - 180 = 0\]

\[= (x - 15)(x + 12)\]

x = 15, x = -12

As the value of x cannot be negative. Hence, the number of sides of a polygon with 90 diagonals is 15.

The number of sides of a polygon with 90 diagonals = 15.


Conclusion

We are provided with the formula to find the number of diagonals of polygons of n sides. The number of diagonals formula, given on this page helps to find the number of diagonals of a polygon of a large number of sides without actually constructing them. So, if you want to find the number of diagonals of any problem, use the formula and get the answer at no time.

FAQs (Frequently Asked Questions)

Q1. What is the General Formula to Find the Number of Diagonals in a Polygon?

Ans. The general formula to find the number of diagonals in a polygon is n(n - 3)/2, where n denotes the number of sides of a polygon.

Q2. What is the Number of Diagonals of a Polygon?

Ans. The diagonals of a polygon are the line segments that connect any two non - adjacent vertices. The number of diagonals of a polygon that can be drawn from each of its vertices is three less than the number of sides or (n - 3). As each diagonal has 2 ends, so this will count the diagonals twice. So to avoid this, we divide by 2 to get the general formula to find the number of diagonals in a polygon i.e. n(n - 3)/2.

Q3. How Many Diagonals Does the Square have?

Ans. The square has two diagonals of equal measurement. The diagonals of a square intersect each other at the centre. The ratio of diagonals of a square to its side is √2.

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