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# Diagonals Formula

A diagonal is a slant line or a sloping line that connects two vertices of a polygon, and those vertices should not be on the same edge. The word diagonal is derived from the Greek word “diagnosis” which means “from angle to angle”. Let us discuss diagonals for different polygons such as square, rectangle, rhombus, parallelogram, etc. with its formulas.

## What Is Diagonals Formula?

The diagonal formula is defined as the formula for diagonals, to find the number of diagonals of different polygons and to find their lengths. The number of diagonal lines of an n-sided polygon = n(n-3)/2 where n is the number of sides

Let's learn about formulas for diagonals of different polygons.

### Formula for Diagonals

The formulas for Diagonals of different polygons can be expressed as,

**Diagonal of Square Formula: **Square Diagonal: a√2

Where a is the length of the side of the square

**Diagonal of Rectangle Formula: **Rectangle Diagonal: √[l^{2} + b^{2}]

Where,

- l is the length of the rectangle.
- b is the breadth of the rectangle.

**Diagonal of Rhombus Formula: **Diagonals of a Rhombus, p = 2(A)/q and q = 2(A)/p

Where

- A is the area of the rhombus
- p and q are the two diagonals of the rhombus.

**Diagonal of Parallelogram Formula: **Parallelogram Diagonals, p=√x^{2}+y^{2}−2xycosA=√x^{2}+y^{2}+2xycosB, q=√x^{2}+y^{2}+2xycosA=√x^{2}+y^{2}−2xycosB

Also, p^{2}+q^{2}=2(a^{2}+b^{2})

Where

- x and y are the sides of the parallelogram
- p and q are the two diagonals of the parallelogram

## Examples Using Diagonals Formula

**Example 1: **Sam is walking around a rectangular park whose length of 10m and breadth is 8m. Find out the diagonal of a rectangular park where Sam is walking?

**Solution:**

To find: The diagonal of a rectangular park.

Given parameters are,

Length = 100m

Breadth = 91m

Using Formula for Diagonals,

Rectangle Diagonal = √[l^{2} + b^{2}]

= √[10^{2} + 8^{2} ]

= √[164]

= 12.80 m

**Answer: The diagonal of a rectangular park where Sam is walking is 12.80 m.**

**Example 2: **The area of the rhombus is 100 inch^{2}. Determine the second diagonal of a rhombus whose one of its diagonal measures is 10 inches.

**Solution: **

To find: The second diagonal of a rhombus

Given parameters are,

The area of the rhombus = 100 inch^{2}

Using the Formula for Diagonals,

Diagonal of a Rhombus, p = 2(A)/q and q = 2(A)/p

p = 2(100)/10

p = 20 inch.

**Answer: The second diagonal of a rhombus is 20 inches.**

Example 3: Determine the length of the diagonal of a square whose side measure is 5 units

**Solution:**

To find: The diagonal of a square

Given parameters are,

Side of square = 5 units

Using Formula for Diagonals,

Square Diagonal = a√2

= 5√2

= 7.07 units

**Answer: The diagonal of a square is 7.07 units.**

## FAQs on Diagonals Formula

### What Is the Formula for Finding Diagonals?

The diagonal formula is used to find the number of diagonals of different polygons and to find their lengths. The formula for finding the number of diagonal of an n-sided polygon = n(n-3)/2 where n is the number of sides whereas for finding the measure of diagonals of different polygonal we have different diagonal formulas.

- Square Diagonal: a√2 (Here a is the length of the side of the square)
- Rectangle Diagonal: √[l
^{2}+ b^{2}] (Here l and b are the length and breadth of the rectangle) - Diagonal of a Rhombus, p = 2(A)/q and q = 2(A)/p (Here A is the area, p and q are the two diagonals of the rhombus)
- Parallelogram Diagonal, p=√x
^{2}+y^{2}−2xycosA = √x^{2}+y^{2}+2xycosB, q=√x^{2}+y^{2}+2xycosA=√x^{2}+y^{2}−2xycosB (Here x and y are the sides, p and q are the two diagonals of the parallelogram)

### How To Use Diagonal Formula?

We use the diagonals formula for calculating the length of the diagonals of a polygon

- Step 1: Identify the polygon.
- Step 2: Check for the number of sides and other given parameters.
- Step 3: Apply the formula for diagonals accordingly.

For finding the length of the diagonals of a square, apply the formula, a√2 where a refers to the length of the side of the square. For finding the length of the diagonals of a rectangle, apply the formula, √[l^{2} + b^{2}] where l and b refer to the length and breadth of the rectangle. For finding the length of the diagonals of a rhombus, apply the formulas, p = 2(A)/q and q = 2(A)/p where A refers to the area, p and q are the two diagonals of the rhombus. For finding the length of the diagonals of a parallelogram, apply the formulas, p=√x^{2}+y^{2}−2xycosA=√x^{2}+y^{2}+2xycosB, q=√x^{2}+y^{2}+2xycosA=√x^{2}+y^{2}−2xycosB where x and y refer to the sides, p and q are the two diagonals of the parallelogram).

### How To Calculate the Number of Diagonals Using Diagonal Formula?

The formula to calculate the number of diagonal of an n-sided polygon = n(n-3)/2 where n is the number of sides of the polygon.

### What Is the Diagonal Formula for Rhombus?

The diagonal formula for rhombus, p = 2(A)/q and q = 2(A)/p, where A is the area of the rhombus whereas p and q are the two diagonals of the rhombus.

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