In this article, we will discuss diagonals and Diagonal Formula in detail. A diagonal line connects two vertices of a polygon and whose vertices are not on the same edge. Also, a diagonal is defined as a sloping line or the slant line. The word diagonal is derived from the Greek word “diagnosis”, which means “from angle to angle”.

The diagonal formula is a formula for calculating the number of diagonals in various polygons and their lengths. An n-sided polygon’s number of diagonal lines = n(n-3)/2, where n is the number of sides. Here we will also discuss the meaning of the diagonal line, diagonals for several polygons such as square, rectangle, rhombus, parallelogram, etc., with its formulas. Continue reading to know more.

What is a Diagonal?

A diagonal line segment connects the two vertices of a shape, which are not already joined by an edge. In other words, a diagonal is a straight line that touches the opposite corners or opposite angles of a polygon through its vertex.

Diagonals of Polygons

A diagonal of a polygon is a line segment obtained by joining any two opposite angles or non-adjacent vertices. We know that a polygon is a closed shape figure formed by joining the adjoining vertices. For example, a triangle has \(3\) sides, a square has \(4\) sides, a pentagon has \(5\) sides, and a hexagon has \(6\) sides, and so on. Depending upon the polygon type, based on the number of edges, the number of diagonals and their properties would differ. Similarly, the properties of diagonals differ according to the solid.

Formula to Find the Number of Diagonals

This formula can be used to calculate the number of diagonals in a polygon. It varies according to the type of polygon, based on the number of sides.
If the number of sides of the polygon is \(n\) then the number of diagonals that can be drawn is given by \(\frac{{n\left( {n – 3} \right)}}{2}\)

We will understand the diagonals of different polygons and some solid shapes as listed below:
1. Diagonal of a Triangle
2. Diagonal of a Square
3. Diagonal of a Rectangle
4. Diagonal of a Parallelogram
5. Diagonal of a Rhombus
6. Diagonal of a Pentagon
7. Diagonal of a Hexagon
8. Diagonal of a Cube
9. Diagonal of a Cuboid

Diagonal of a Triangle

A triangle is a closed shape with \(3\) sides, \(3\) angles, and \(3\) vertices. A triangle is the primary type of polygon. No vertices in a triangle are non-adjacent. This means that there are no line segments that can form diagonals joining any two vertices.
Therefore, the number of diagonals of a triangle \(=0.\)

Diagonal Formula of a Square

A closed two-dimensional figure having four sides and four corners is known as a square. All the sides of a square are parallel to each other with equal lengths. The diagonal of a square is a line segment that joins any two of its opposite angles. In the below-shown square, there are two pairs of non-adjacent vertices. By joining the opposite angles of each such pair, we get two diagonals, \(AC\) and \(BD\) of the square.

The lengths of the lines \(AC\) and \(BD\) in the below square are the same. The diagonal of any square can be divided into two equal right-angled triangles, such that the diagonal makes hypotenuse of the right-angled triangle so formed.
Therefore, the number of diagonals of a square \(=2.\)

The formula for the diagonal of a square with the length of a side a is given by: \(d = \sqrt 2 a\)

Diagonal Formula of a Rectangle

A line segment that joins any two of its non-adjacent vertices or opposite angles is the diagonal of a rectangle. In the below rectangle \(AC\) and \(BD\) are the diagonals. We can notice that the lengths of both \(AC\) and \(BD\) are the same. A diagonal divides a rectangle into \(2\) right-angled triangles, in which the sides are the same as the sides of the rectangle and with a hypotenuse. That hypotenuse is the diagonal of the given rectangle.

The formula for the diagonal of a rectangle with length \(l\) and breadth \(b\) is given by \(d = \sqrt {{l^2} + {b^2}} .\)

Diagonal Formula of a Parallelogram

A parallelogram is also a quadrilateral. The opposite sides and angles of a parallelogram are equal, and the diagonals bisect each other. In the below figure, \(AC\) and \(BD\) are the diagonals of a parallelogram.

The length of the diagonals of the parallelogram is calculated using the formula:
Diagonal \({d_1} = p = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,A} = \sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,B} \)
Diagonal \({d_2} = q = \sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,A} = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,B} \)
\( \Rightarrow {p^2} + {q^2} = 2\left( {{a^2} + {b^2}} \right)\)

Diagonal Formula of a Rhombus

The line segments joining the opposite angles or opposite vertices, bisecting each other at a \({90^{\rm{o}}}\) angle, are the rhombus’ diagonals. The two halves of any diagonal will be of the same opposite length. A rhombus can be defined as a kite-shaped quadrilateral having all four sides equal. The diagonals of a rhombus will have non-identical values, except the rhombus is a square.
In the below figure, \(AC\) and \(BD\) are the diagonals of a Rhombus.

If \(p\) and \(q\) are the lengths of the diagonal of a rhombus, then the formula to find the diagonal of a rhombus is given by \(p = 2\frac{A}{q}\) where \(A\) is the area of the rhombus.

Diagonal Formula of a Pentagon

A two-dimensional figure having five sides and five corners are known as a pentagon. The measure of all five sides is equal in a regular pentagon. There are five diagonals in a pentagon, as shown in the figure below.

If \(a\) is the side of a pentagon, then the formula to find the diagonal of a regular pentagon is given by \(d = \frac{{1 + \sqrt 5 }}{2}a\)

Diagonal Formula of a Hexagon

A two-dimensional figure with six sides and six corners is known as a hexagon. The length of all sides is equal in a regular hexagon. There are nine diagonals in a hexagon, as shown in the image below.

Diagonal Formula of a Cube

A three-dimensional solid that has edges of all the same length is known as a cube. That means that the length, breadth, and height are equal, and each of its faces is a square. The principal diagonal is the line segment that divides through its centre, joining the opposite vertices. At the same time, the diagonal of a cube’s face is the one joining the opposite vertices on every face. Thus, it is not the main diagonal.

The diagonal of a cube with the length of the side \(a\) units is given by: \(d = \sqrt 3 a\)

Diagonal Formula of a Cuboid

A three-dimensional solid shape with six rectangular surfaces or four rectangular and two square surfaces is known as a cuboid. The principal diagonal of a cuboid is the one that divides through the centre of the cuboid.

The formula for the diagonal of a cuboid with length \(l,\) breadth \(b\) and height \(h\) is given by: \(d = \sqrt {{l^2} + {b^2} + {h^2}} \)

Solved Examples– Diagonal Formula

Q.1. How many sides does a polygon of \(90\) diagonals have?
Ans: We know that if the number of sides of the polygon is \(n\) then the number of diagonals that can be drawn is given by \(\frac{{n\left( {n – 3} \right)}}{2}\)
\( \Rightarrow \frac{{n\left( {n – 3} \right)}}{2} = 90\)
\( \Rightarrow n\left( {n – 3} \right) = 180\)
\( \Rightarrow {n^2} – 3n = 180\)
\( \Rightarrow {n^2} – 3n – 180 = 0\)
\( \Rightarrow \left( {n – 15} \right)\left( {n + 12} \right) = 0\)
\( \Rightarrow n = 15\)
Therefore, the number of sides of the given polygon is \(15.\)

Q.2. Find the diagonal of a cube with the given side \(4\,{\rm{cm}}.\)
Ans: We know that the diagonal of a cube with the length of the side \(a\) units is given by: \(d = \sqrt 3 a\)
Given: \(a = 4\,{\rm{cm}}\)
So, \(d = \sqrt 3 \times 4\,{\rm{cm}}\)
\( \Rightarrow d = 1.734 \times 4\,{\rm{cm}}\)
\( \Rightarrow d = 6.936\,{\rm{cm}}\)
Therefore, the length of the diagonal of the cube is \(6.936\,{\rm{cm}}\)

Q.3. If the length, breadth and height of a cuboid are \(12\,{\rm{cm}},\,9\,{\rm{cm}}\) and \(8\,{\rm{cm}}\) respectively. What is the measure of the diagonal of a cuboid?
Ans: We know that the formula for the diagonal of a cuboid with length \(l,\) breadth \(b\) and height \(h\) is given by:
\(d = \sqrt {{l^2} + {b^2} + {h^2}} \)
Given: \(l = 12\,{\rm{cm}},\,b = 9\,{\rm{cm}}\) and \(h = 8\,{\rm{cm}}\)
\( \Rightarrow d = \sqrt {{{12}^2} + {9^2} + {8^2}} \)
\( \Rightarrow d = \sqrt {144 + 81 + 64} \)
\( \Rightarrow d = \sqrt {289} \)
\( \Rightarrow d = 17\,{\rm{cm}}\)
Therefore, the length of the diagonal is \(17\,{\rm{cm}}.\)

Q.4. Find the length of the diagonal of a square whose side is \(5\,{\rm{cm}}.\)
Ans: We know that the formula for the diagonal of a square with the length of a side \(a\) is given by \(d = \sqrt 2 a.\)
Given: \(a = 5\,{\rm{cm}}\)
So, \(d = \sqrt 2 \times 5\,{\rm{cm}}\)
Therefore, the length of the diagonal of the given square is \(5\sqrt 2 \,{\rm{cm}}{\rm{.}}\)

Q.5. Find the length of each diagonal of a rectangle of length \(4\) units and width \(6\) units.
Ans: We know that the formula for the diagonal of a rectangle with length \(l\) and breadth \(b\) is given by \(d = \sqrt {{l^2} + {b^2}} \)
Given: \(l = 4\) units and \(b = 6\) units
So, \(d = \sqrt {{4^2} + {6^2}} \)
\( \Rightarrow d = \sqrt {16 + 36} \)
\( \Rightarrow d = \sqrt {52} \) units.
Therefore, the length of the diagonal of the given square is \(\sqrt {52} \) units.

Summary

In the above article, we have studied the definition of diagonal and the formula to find the diagonals of a rectangle, square, pentagon, parallelogram, hexagon, rhombus and three-dimensional shapes like cube and cuboid. Also, we have solved some example problems on the diagonal formula.

Frequently Asked Questions (FAQs)– Diagonal Formula

Q.1. What is the formula of the diagonal of a square?
Ans: The formula for the diagonal of a square with the length of a side \(a\) is given by \(d = \sqrt 2 a.\)

Q.2. What is the formula of the diagonal of the cube?
Ans: The diagonal of a cube with the length of the side \(a\) units is given by: \(d = \sqrt 3 a.\)

Q.3. How many diagonals does a decagon have?
Ans: The number of sides of a decagon is \(10.\)
So, the number of diagonals can be calculated by: \(\frac{{10\left( {10 – 3} \right)}}{2} = \frac{{70}}{2} = 35\)
Therefore, we find \(35\) diagonals in a decagon.

Q.4. What is the relationship between the diagonals of a trapezium?
Ans: The diagonals of isosceles trapezium bisect each other. The length of the mid-segment is the same as half the sum of the parallel bases in a trapezium. Two pairs of adjoining angles of a trapezium formed between the parallel sides, and one of the non-parallel sides, are supplementary.

Q.5. How to find the number of diagonals in a regular polygon?
Ans: If the number of sides of the polygon is \(n\) then the number of diagonals that can be drawn is given by \(\frac{{n\left( {n – 3} \right)}}{2}.\)

We hope you find this detailed article on the diagonal formula helpful. If you have any doubts or queries regarding this topic, feel to ask us in the comment section and we will assist you at the earliest. Happy learning!