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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:

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

Where the number of sides of a polygon is n

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

Let us study diagonal formula for different shapes

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

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

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.

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

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.