### Math

# Diagonal Formula with Problem Solution & Solved Example

Table of Contents

The diagonal formula in mathematics is used to calculate the diagonals of a polygon including rectangles, square, and more similar shapes. When two non-adjacent vertices within a polygon are joined through a single line, it is named as the polygon. Diagonal is formed by joining any two vertices of a polygon except edges. The sloping is also named as the diagonal. Here, we will discuss for diagonal calculation in case of rectangle and the square.

Square is nothing but a regular quadrilateral whose four sides are equal and aligned at the angle of 90 degrees. This is the reason diagonals for a square are also equal. In the case of rectangle, the opposite sides are parallel and they are congruent too. In this case, the diagonals of a rectangle will bisect each other and they are congruent too. Below is give the Diagonal Formula for square and the rectangle.

## Diagonal Formula

#### Diagonal Of Square Formula

\[\LARGE Diagonal\;of\;a\;Square=a\sqrt{2}\]

Where, a is the length of the side of the square

#### Diagonal Of Rectangle Formula

\[\LARGE Diagonal\;of\;a\;Rectangle=\sqrt{l^{2}+b^{2}}\]

Where,

l is the length of the rectangle.

b is the breadth of the rectangle.

p and q are the diagonals

#### Diagonal Of A Cube Formula

\[\LARGE Diagonal\;of\;a\;Cube=\sqrt{3}x\]

Where,x is the length of the side of the Cube

#### Diagonal Of Polygon Formula

**Polygon formula to find area:**

\[\large Area\;of\;a\;regular\;polygon=\frac{1}{2}n\; sin\left(\frac{360^{\circ}}{n}\right)s^{2}\]

**Polygon formula to find interior angles:**

\[\large Interior\;angle\;of\;a\;regular\;polygon=\left(n-2\right)180^{\circ}\]

**Polygon formula to find the triangles:**

\[\large Interior\;of\;triangles\;in\;a\;polygon=\left(n-2\right)\]

Where,** n** is the number of sides and **S** is the length from center to corner.

#### Diagonal Of Parallelogram Formula

Formula of parallelogram diagonal in terms of sides and cosine $\beta$ (cosine theorem)

\[\LARGE p=d_{1}=\sqrt{a^{2}+b^{2}- 2ab\;cos \beta}\]

\[\LARGE q=d_{2}=\sqrt{a^{2}+b^{2}+ 2ab\; cos \beta}\]

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

\[\LARGE p=d_{1}=\sqrt{a^{2}+b^{2}+2ab\;cos \alpha }\]

\[\LARGE q=d_{2}=\sqrt{a^{2}+b^{2}-2ab\;cos\alpha }\]

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

\[\LARGE p=d_{1}=\sqrt{2a^{2}+2b^{2}-d_{2}^{2}}\]

\[\LARGE q=d_{2}=\sqrt{2a^{2}+2b^{2}-d_{1}^{2}}\]

To find the total number of diagonals for a polygon with n number of sides, you can use the following formula. This formula is applicable to all shapes that satisfy the properties of a Polygon. No formula in mathematics is written in dreams but there is the logic behind it and the same is true for the below-given formula as well. Just memorize it by heart and start solving tough problems in minutes.