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# Diagonal Formula with Problem Solution & Solved Example

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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.