Diagonal Formula

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What is Diagonal?

We have studied what is a polygon. To recall, a two-dimensional closed figure bounded with three or more than three straight lines is called a polygon. Triangles, square, rectangle, pentagon, hexagon, are some examples of polygons.


The segments are referred to as the sides of the polygon. The points at which the segments meet are called vertices. Segments that share a vertex are called adjacent sides. A segment whose endpoints are nonadjacent vertices is called a diagonal.


A diagonal of a polygon is a line from a vertex to a non-adjacent vertex. So a triangle has no diagonals. You cannot draw a line from one interior angle to any other interior angle that is not also a side of the triangle. A quadrilateral, the next-simplest, has two diagonals. A pentagon has five diagonals. The shapes with a diagonal drawn on them are given below: 


(image will be uploaded soon)


We have a number of diagonals formula of a polygon which help us to know the number of diagonals in a polygon that is 


\[\text{Diagonal of Polygon} = \frac{n(n-3)}{2}\]


Where the number of sides of a polygon is n


From this formula, we can easily calculate the number of diagonals in a polygon.


The Given Table Shows the Number of Diagonals in Different Polygons

Shape Names

Number of Vertices

Number of Diagonals

Triangle

3

0

Quadrilateral

4

2

Pentagon

5

5

Hexagon

6

9

Septagon

7

14

Octagon

8

20

Nonagon

9

27

Decagon

10

35


Let us study diagonal formula for different shapes


Diagonal of Square Formula

\[\text{Diagonal of Square Formula} = s \sqrt{2}\]


Where, s represents the length of the side of the square


Diagonal of Rectangle Formula

\[\text{Diagonal of Rectangle Formula} = \sqrt{(a^{2} + b^{2})}\]


Where,

a is the length of the rectangle.

b is the breadth of the rectangle.

p and q are the diagonals.


Diagonal of Parallelogram Formula

The formula of parallelogram diagonal in terms of sides and cosine β (cosine theorem) if x =d1 and y = d2 are the diagonals of a parallelogram and a and b are the two sides.


\[x = d_{1} = \sqrt{(a^{2} + b^{2} - 2 ab cos \beta})\]


\[y = d_{2} = \sqrt{(a^{2} + b^{2} + 2 ab cos \beta})\]


The formula of parallelogram diagonal in terms of sides and cosine α (cosine theorem)


\[x = d_{1} = \sqrt{(a^{2} + b^{2} + 2 ab cos \alpha})\]


\[y = d_{2} = \sqrt{(a^{2} + b^{2} - 2 ab cos \alpha})\]


Formula of diagonal of parallelogram in terms of two sides and other diagonal


\[x = d_{1} = (\sqrt{2a^{2} + 2b^{2} - d_{2}^{2}})\]


\[y = d_{2} = (\sqrt{2a^{2} + 2b^{2} - d_{1}^{2}})\]


Diagonal of a Cube Formula

For a cube, we find the diagonal by using a three-dimensional version of the Pythagorean Theorem/distance formula:


\[\text{The formula of diagonal of cube} = s\sqrt{3}\]


Where s is the side of a cube.


Diagonal of Cuboid Formula

\[\text{Diagonal of Cuboid Formula} = \sqrt{(l^{2} + b^{2} + h^{2})}\]


Where l, b, and h are length, breadth, and height of a cuboid


Solved Examples

Example 1: Find the diagonal of a rectangle with length as 8 cm and breadth value as 6 cm.


Solution: As given,

l = 8 cm

b= 6 cm

Formula for diagonal of rectangle is,

\[d =\sqrt{l^{2} + b^{2}}\]

Substituting the values,

\[d =\sqrt{8^{2} + 6^{2}}\]

\[d =\sqrt{(64 + 36)\]

\[d = \sqrt{100}\]

d = 10 cm

Therefore, diagonal will be 10 cm.


Example 2: Find the length of the diagonal of a cube with side length as 10 cm?


Solution: As given,

a = 10 cm

The formula for diagonal of a cube is,

\[d = a\sqrt{3}\]

substituting the values,

\[d = 10 \times \sqrt{3}\]

i.e. d = 10 X 1.73

i.e.d = 17.3 cm

Therefore, diagonal will be 17.3 cm.


Quiz Time

  1. A rectangle has a length of 14 cm and a width of 10cm. What is the diagonal of the rectangle?

  2. How many diagonals does an eighteen-sided polygon have?

FAQ (Frequently Asked Questions)

1. State Diagonals Examples in Real Life.

Answer: Some of the real-life examples of diagonals are:

  • Diagonals in squares and rectangles add strength to construction, whether for a house wall, bridge, or tall building. 

  • You may see diagonal wires used to keep bridges steady. 

  • When houses are being built, look for diagonal braces that hold the walls straight and true.

  • Bookshelves and scaffolding are braced with diagonals.

  • For a catcher in softball or baseball to throw out a runner at second base, the catcher throws along a diagonal from home plate to second.

  • The phone or computer screen is measured along its diagonal.

2. What is the Formula to Find the Number of Diagonals in any Polygon?

Answer: Formula for the number of diagonals

As we know that, the number of diagonals from a single vertex is three less than the number of vertices or sides, or (n-3).


Consider there are n number of sides and n vertices, which gives us n(n-3) diagonals. Each diagonal has two ends, so this would count each diagonal twice. So we divide by 2, for the final formula:


Number of diagonals = n(n−3)/ 2

Where n is the number of sides.