A polygon is a mathematical figure surrounded by straight lines. Poly means numerous in Greek, while gon indicates angle. The number of diagonals in a polygon is calculated using the diagonal of a polygon formula. The most basic polygon is a triangle with three sides and three angles summing 180 degrees.

This article will discuss the definition of a polygon, diagonal, types of polygons, and formula to find the number of diagonals in a polygon.

What is a Polygon?

Any two-dimensional (flat) object with only straight sides that close in a space and no sides cross each other is called a simple polygon (if they do, it is a complex polygon). A polygon is a triangle. Polygons include the dart, kite, quadrilateral, and star. Concave and convex polygons are two types of simple polygons.

What is a Diagonal?

A line connecting a vertex to a non-adjacent vertex is referred to as a polygon’s diagonal. The simplest polygon, a triangle, has no diagonals. We cannot draw a line from one interior angle to any other interior angle of the triangle. A quadrilateral has two diagonals, making it the next-simplest shape. There are five diagonals in a pentagon, whether regular or irregular and nine diagonals in a hexagon.

Diagonals will always be within the interior of convex, simple polygons. Consider the shape of a rectangular door. A line can be drawn from the top hinge corner to the bottom hinge corner on the opposite side. A line from the bottom hinge corner to the top of the opposite corner can also be drawn. There will be two such diagonals formed.

The diagonals of concave, simple polygons may extend beyond the polygon, crossing sides and partially lying in the shape’s exterior. They’re still diagonals, though. Concave polygons with diagonals beyond their forms are common in darts and stars.

Diagonal of a Polygon Formula

We can simply count all of the possible diagonals of a basic polygon with a few sides. Counting polygons may be complicated when they become more complex.

Fortunately, there is a simple formula for determining how many diagonals a polygon contains. Because each vertex (corner) is connected to two other vertices via sides, those connections cannot be considered diagonals. That vertex, too, is unable to link to itself.

So, for \(n\) sides, we’ll subtract the number of diagonals available by three. We don’t want to count the same diagonal twice, either. So, the result will have to be divided by two.
Or this formula is easily generated by subtracting the total sides from the combination of diagonals that each vertex sends to another vertex. To put it another way, an \(n\)-sided polygon contains \(n\)-vertices that can be connected in \(^n{C_2}.\) ways
The formula derived by subtracting \(n\) from \(^n{C_2}\) methods \(^n{C_2} – n = \frac{{n\left({n – 1} \right)}}{2} – n\)
\( = \frac{{n\left({n – 3} \right)}}{2}.\)

For example, the quadrilateral only has two diagonals if you don’t include the adjacent two sides and the vertex too is unable to link to itself.

Hence, for an \(n\)-sided regular polygon, the number of diagonals can be obtained using the formula given below:
Number of diagonals \( = \frac{{n\left({n – 3} \right)}}{2}\)
For a pentagon, the number of vertices is \(n = 5\)
So the number of diagonals can be calculated by using the above formula, we get
Number of diagonals in a pentagon \( = \frac{{5\left({5 – 3} \right)}}{2} = \frac{{5 \times 2}}{2} = 5\)

Length of Diagonal of Rectangle, Square and Cube

Diagonal of a Rectangle

If \(l\) is the length of the rectangle, and \(b\) is the breadth of the rectangle.
Length of diagonal of a rectangle, \(d = \sqrt {{l^2} + {b^2}} \)

Diagonal of a Square

Now let’s look at a few different diagonal formulas to find the length of a diagonal.

Diagonal of \(a\) square, \(d = a\sqrt 2 \)
Where \(a\) is the side of the square.

Diagonal of a Cube

For a cube, we find the diagonal by using a three-dimensional version of the Pythagoras theorem.

Diagonal of a cube, \(d = \sqrt {{a^2} + {a^2} + {a^2}} = \sqrt {3a} \)
Where \(a\) is the side of the cube.

Diagonals in our Daily Life

Diagonals in rectangles, as well as diagonals in squares, provide strength to a building’s structure, whether it’s a home wall, a bridge, or a towering structure. You may have noticed diagonal cables used to stabilise the bridges. Look for diagonal bracing in the door’s construction to keep the door straight and true.

Diagonal braces are used to brace bookshelves and scaffolding. When a catcher throws out a runner at second base in softball or baseball, the catcher throws across the diagonal from home plate to second base.

The diagonal of the phone or computer screen on which you are watching this course is measured. The width and height of a \(21”\) screen are never specified; it is \(21”\) from one corner to the other corner.

Solved examples

Q.1. In a \(20\)-sided polygon, one vertex does not send any diagonals. Find out how many diagonals does that \(20\)-sided polygon contain.
Ans: We know that, number of diagonals in a polygon \( = \frac{{n\left({n – 3}\right)}}{2}.\)
In a \(20\)-sided polygon, the total diagonals are \( = \frac{{20\left({20 – 3} \right)}}{2} = 170.\)
But, since one vertex does not send any diagonals, the diagonals by that vertex must be subtracted from the total number of diagonals.
In a polygon, it is known that each vertex makes \(\left({n – 3} \right)\) diagonals. In this polygon, each vertex makes \(\left({20 – 3} \right) = 17\) diagonals.
Since \(1\) vertex does not send any diagonal, the total diagonal in this polygon will be \(\left({170 – 17} \right) = 153\) diagonals.

Q.2 In an \(11\)-sided regular polygon, find the total number of diagonals.
Ans: Method I: In an \(11\)-sided polygon, total vertices are \(11.\) Now, the \(11\) vertices can be joined with each other by \(^{11}{C_2} = \frac{{11 \times 10}}{2} = 55\) ways.
Now, there are \(55\) diagonals possible for an \(11\)-sided polygon which includes its sides also. So, subtracting the sides will give the total diagonals contained by the polygon.
So, total diagonals contained within an \(11\)-sided polygon \( = 55 – 11 = 44.\)
Method II: According to the formula, the number of diagonals in a polygon \( = \frac{{n\left({n – 3} \right)}}{2}.\)
So, an \(11\)-sided polygon will contain \( = \frac{{11\left({11 – 3} \right)}}{2} = 44\) diagonals.

Q.3. How many sides does a polygon have if it has \(90\) diagonals?
Ans: Suppose that the number of sides of the given polygon is \(n.\)
The number of diagonals \( = 90.\)
We know that, number of diagonals in a polygon \( = \frac{{n\left({n – 3} \right)}}{2}\)
Where \(n\) is the number of sides of the polygon
\( \frac{{n\left({n – 3} \right)}}{2} = 90\)
\(n\left({n – 3} \right) = 180\)
\({n^2} – 3n – 180 = 0\)
\(\left({n – 15} \right)\left({n + 12} \right) = 0\)
\(n = 15;n = – 12\)
Since sides cannot be negative, the value of \(n\) is \(15.\)

Q.4 One vertex in a \(12\)-sided polygon does not have any diagonals. Calculate the number of diagonals in the \(12\)-sided polygon.
Ans: We know that,
Number of diagonals in a polygon \( = \frac{{n\left({n – 3} \right)}}{2}\)
Where \(n\) is the number of sides of the polygon
In a \(12\)-sided polygon, \(n = 12\)
Number of diagonals \( = \frac{{12\left({12 – 3} \right)}}{2}\)
\( = 54\)
Now, it is given that one vertex does not have any diagonal, so we need to subtract the number of diagonals of that vertex from the total number of diagonals.
In a polygon, each vertex makes \(\left({n – 3} \right)\) diagonals, therefore in a \(12\)-sided polygon, each vertex makes \(\left({12 – 3} \right) = 9\) diagonals
Hence, the total number of diagonals in this polygon is \( = \left({54 – 9} \right) = 45\)

Q.5. A polygon has \(27\) diagonals. Identify the number of sides present in the polygon.
Ans: We know that,
Number of diagonals in a polygon \( = \frac{{n\left({n – 3} \right)}}{2}\)
Where \(n\) is the number of sides of the polygon.
Now, according to the given question,
\(\frac{{n\left({n – 3} \right)}}{2} = 27\)
\(n\left({n – 3} \right) = 54\)
\({n^2} – 3n – 54 = 0\)
\({n^2} – 9n + 6n – 54 = 0\)
\(n\left({n – 9} \right) – 6\left({n – 9} \right) = 0\)
\(\left({n + 6} \right)\left({n – 9} \right) = 0\)
\(n = 9, – 6\)
Since \(n\) is a negative value, the number of sides present in the polygon is \(9.\)

Summary

We have learnt a lot about the diagonals of polygons, which are essential. We now know how to get the diagonals of any polygon and some real-world instances and apply the method. We also went through how to find the diagonal length in cubes, squares, and rectangles using diagonal formulas.

FAQs

Q.1. What is the polygon formula?
Ans: The polygon formula for finding the number of diagonals is \(n\frac{{\left({n – 3} \right)}}{2}\)
Where \(n\) is the number of sides of the polygon.

Q.2. How do you find the number of sides of a polygon when the number of diagonals is given?
Ans: If you know how many diagonals a polygon has, you can calculate how many sides it uses the formula for the number of diagonals of a polygon with \(n\) sides. Find the number of sides in a polygon with \(54\) diagonals, for example.
Number of diagonals in a polygon \( = n\frac{{\left({n – 3} \right)}}{2}\)
Where \(n\) is the number of sides of the polygon.
Given, number of diagonals \( = 54\)
\(54 = \frac{{n\left({n – 3} \right)}}{2}\)
\(54 \times 2 = 108 = n \times \left({n – 3} \right)\)
\(108 = {n^2} – 3n\)
This is a quadratic equation, so it can be solved either by factoring or using the quadratic formula.
\(0 = {n^2} – 3n – 108\)
\(0 = \left({n – 12} \right)\left({n + 9} \right)\)
So, \(n = 12\) or \( – 9.\)
We know that this polygon must have \(12\) sides since a polygon cannot have a negative number of sides.

Q.3. What is a diagonal of a polygon?
Ans: A diagonal of a polygon is a line segment formed by joining any two non-adjacent vertices.

Q.4 Which way is the diagonal line?
Ans: A diagonal is defined as a line from one corner to the farthest corner.

Q.5. What is the example of a diagonal line?
Ans: A line drawn from the bottom left corner of a square to the top right corner is an example of a diagonal. Diagonal braces are used to brace bookshelves and scaffolding.

We hope this detailed article on the diagonal of a polygon formula helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!