Table of Contents

**Diagonal of a Polygon Formula**

**Introduction**

Diagonals are the parts of a shape, in geometry. In Mathematics, a diagonal is a line that connects two vertices of a polygon or a solid, whose vertices are not on the same edge. In general, a diagonal is defined as a sloping line or the slant line, that connects to the vertices of a shape. Diagonals are defined as lateral shapes that have sides/edges and corners. We can find the diagonals for curved shapes, such as circles, spheres, cones, etc.

The word diagonal is derived from the Greek word “diagonios” which means “from angle to angle”. Also, in matrix algebra, the diagonal of the square matrix defines the set of entities from one corner to the farthest corner. In this article, let us discuss the meaning of the diagonal line, diagonals for different polygons such as square, rectangle, rhombus, parallelogram, etc. with its formulas. Before going to learn the diagonal of a polygon formula, let us recall what is a polygon and what is a diagonal. A polygon is a closed shape made with 3 or more line segments, A diagonal of a polygon is a line segment that is obtained by joining any two non-adjacent vertices. Let us learn the diagonal of a polygon formula along with a few solved examples.

**What Is the Diagonal of a Polygon Formula?**

The diagonal of a polygon formula is used to calculate the number of diagonals of a polygon. It says

The number of diagonals of a polygon = n(n−3)/2

Here

‘n’ is the number of sides polygon has.

Let us see the applications of the diagonal of a polygon formula in the following section.

**Solved Examples Using Diagonal of a Polygon Formula**

**Example 1:** Find the number of diagonals of a decagon using the diagonal of a polygon formula.

Solution:

The number of sides of a decagon is, n=10

The number of diagonals of a decagon is calculated using:

n(n−3)/2 = 10(10−3)/2

=10(7)/2

=70/2

=35

The number of diagonals of a decagon= 35.

**Example 2:** If a polygon has 90 diagonals, how many sides does it have?

Solution:

Let us assume that the number of sides of the given polygon is n.

The number of diagonals = 90.

Using the diagonal of a polygon formula,

n(n−3)/2 = 90

n(n−3) = 180

n2 − 3n − 180 = 0

(n − 15)(n + 12) = 0

n = 15; n = −12

Since n cannot be negative, the value of n is 15.

Sides of the given polygon = 15.

**Frequently Asked Questions on Diagonals of a Polygon Formula**

1: What is diagonal?

Answer: A diagonal is a straight line that connects the opposite corners of the polygon through its vertex.

2: What is the formula to calculate the diagonal of a polygon?

Answer: If “n” is the number of vertices of a polygon, the formula to calculate the number of diagonals of a polygon is [n(n-3)]/2

3: What is the length of the diagonal of a square?

Answer: The length of the diagonal of the square is a√2, where “a” is the length of any side of a square.

4: How many diagonals does a heptagon have?

Answer: A heptagon has 14 diagonals, as it has 7 vertices.

5: Does a circle have a diagonal?

Answer: No, a circle does not have a diagonal, as it has no vertices and sides.

6: What is the full form of polygon?

Answer: Polygon is the combination of two words, i.e. poly (means many) and gon (means sides).

7: What are the Different Types of Polygons?

Answer: There are different types of polygons that are named according to the number of sides that they have. For example, a 3-sided polygon is called a triangle, a 4-sided polygon is called a quadrilateral, a 5-sided polygon is called a pentagon, a 6-sided polygon is called a hexagon, a 7-sided polygon is called a heptagon, and so on.

8: What is the Difference Between Concave and Convex Polygons?

Answer: The differences between concave and convex polygons are given below.

Concave polygons are those polygons that have at least one interior angle which is a reflex angle and it points inwards. But a convex polygon has no interior angle that measures more than 180°.

Concave polygons have a minimum of 4 sides but a convex polygon can have 3 sides.

A few of the diagonals in a concave polygon may lie partly or fully outside it. But no diagonal in a convex polygon lies outside it.

It is to be noted that all concave polygons are irregular because the interior angles are not equal. However, a convex polygon may not always be an irregular polygon.