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# Diagonal of Rectangle Formula

Before going to know what is the **diagonal of a rectangle formula**, let us understand what is meant by the diagonal of a rectangle. The diagonal of a rectangle is a line segment that joins any two of its non-adjacent vertices. The formula for the diagonal of a rectangle gives the length of the diagonal when the dimensions of the rectangle are known. Let us learn the diagonal of the rectangle formula along with a few solved examples.

## What is the Diagonal of Rectangle Formula?

The diagonal of a rectangle is a line segment that joins any two of its non-adjacent vertices. In the following rectangle, AC and BD are the diagonals. You can see that the lengths of both AC and BD are the same. A diagonal cuts a rectangle into 2 right triangles, in which the sides are equal to the sides of the rectangle and with a hypotenuse. That hypotenuse is the diagonal. In the following rectangle, AC and BD would be the diagonals. The lengths of both AC and BD are the same.

### Diagonal of a Rectangle Formula

The length of diagonals of a rectangle can be calculated as,

d = √( l^{2} + w^{2})

where,

- l = length of the rectangle
- w = width of the rectangle

## Diagonal of Rectangle Derivation

The diagonal of a rectangle formula is derived using Pythagoras theorem. Let us consider a rectangle of length "l" and width "w". Let the length of each diagonal be "d".

Applying Pythagoras theorem to the triangle ABD,

d^{2} = l^{2} + w^{2}

Taking square root on both sides,

d = √( l^{2} + w^{2})

Thus, the diagonal of a rectangle formula is:

d = √( l^{2} + w^{2})

where,

- l = length of the rectangle
- w = width of the rectangle

Let us see the applications of the diagonal of rectangle formula in the following section.

## Examples Using Diagonal of Rectangle Formula

**Example 1:** Find the length of each diagonal of a rectangle of length 8 units and width 6 units.

**Solution:**

To find: The length of each diagonal of the given rectangle.

It is given that:

The length of the rectangle, l = 8 units.

The width of the rectangle, w = 6 units.

Using the diagonal of a rectangle formula,

d = √( l^{2} + w^{2})

d = √( 8^{2} + 6^{2})

= √100

= 10 units.

**Answer:** The length of each diagonal = 10 units.

**Example 2:** The size of the screen of a television is the length of its diagonal. Then find the size of the television whose dimensions are 16 inches and 40 inches.

**Solution:**

To find: The size (diagonal) of the given television.

It is given that:

The length of the television, l = 40 units.

The width of the television, w = 16 units.

Using the diagonal of a rectangle formula,

d = √( l^{2} + w^{2})

d = √( 40^{2} + 16^{2})

= √1856

= 43.08 inches.

**Answer:** The size (diagonal) of the given television = 43.08 inches.

**Example 3:** The dimensions of a rectangle are 4 units and 3 units. Determine the length of each diagonal of a rectangle.

**Solution:**

To find: The length of each diagonal of the given rectangle.

It is given that:

The length of the rectangle, l = 4 units.

The width of the rectangle, w = 3 units.

Using the diagonal of a rectangle formula,

d = √( l^{2} + w^{2})

d = √( 4^{2} + 3^{2})

= √25

= 5 units.

**Answer:** The length of each diagonal = 5 units.

## FAQs on Diagonal of a Rectangle Formula

### What Is the Diagonal of a Rectangle Formula in Math?

In math, the diagonal of a rectangle formula gives the length of the diagonal when the dimensions of the rectangle are known which include the measure of the two adjacent sides of the rectangle. It is given as,

d = √( l^{2} + w^{2})

where,

- l = length of the rectangle
- w = width of the rectangle

### How To Use the Diagonal of a Rectangle Formula?

The diagonal of a rectangle formula is used to find the diagonal of a rectangle when two of its dimensions are given,

- Step 1: Identify the length and width of the given rectangle.
- Step 2: Put the values in the formula, d = √( l
^{2}+ w^{2})

### What Is 'w' in the Diagonal of a Rectangle Formula?

In the diagonal of a rectangle formula, 'w' refers to the width of the rectangle. The formula is given as, d = √( l^{2} + w^{2})**, **where, l is the length of the rectangle and w is the width of the rectangle.

### How To Derive the Diagonal of a Rectangle Formula?

The diagonal of a rectangle formula can be derived using the Pythagoras theorem.

- Step 1: Consider a rectangle of length "l" and width "w".
- Step 2: Let the length of the diagonal be "d".
- Step 3: After constructing the diagonal, the rectangle gets divided into two right triangles.
- Step 4: Consider any one of the triangles, its two sides will be 'l' and 'w' and the diagonal 'd' of the rectangle will be its hypotenuse.
- Step 5: Applying Pythagoras theorem, d = √( l
^{2}+ w^{2})

### How to Find the Length of Rectangle Using the Diagonal of a Rectangle Formula?

We can find the length of the rectangle using the Diagonal of a Rectangle Formula if the length of the diagonal and the width are given. Then, we can find the length using formula, l = √(d^{2} - w^{2}).

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