## 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.

## 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 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 length of the rectangle and w is 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})

## What’s a Diagonal?

A **diagonal** of a polygon is a line from a vertex to a non-adjacent vertex. So a triangle, the simplest polygon, 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, whether regular or irregular, has five diagonals.

**In convex, simple polygons, diagonals will always be within the interior**. Consider a rectangular door. You can run a line from the top hinge corner to the bottom, opposite corner. You can also run a line from the bottom hinge corner up to the top, opposite corner. Those are the only two diagonals possible.

In **concave, simple polygons, the diagonals may go outside the polygon**, crossing sides and partly lying in the shape’s exterior. They are still diagonals. Darts and stars are typical examples of concave polygons with diagonals outside their shapes.

Do *not* attempt to apply these concepts and our diagonal formula to complex polygons (polygons with self-intersecting lines).

### Diagonals in Real Life

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 you are viewing this lesson on is measured along its diagonal. A 21″ screen never tells you the width and height; it is 21″ from one corner to an opposite corner.

## Diagonal of a Polygon Formula

To find all possible diagonals of a simple polygon with just a few sides, you can easily count them. When the polygon gets a bit complicated, counting them can be very hard.

Fortunately, an easy formula exists to tell you exactly how many diagonals a polygon has. Remember that any vertex (corner) is connected by sides to two other vertices, so those connections cannot count as diagonals. That vertex cannot connect to itself, either. So for n sides, we will immediately reduce the possible number of diagonals by three.

We also do not want to count the same diagonal twice. Our door, for example, only has two diagonals; you do not count going from the top hinge to bottom opposite and back again. You will have to divide any answer by two.

### Diagonal Formula

This leaves us with an elegant formula, where n is the number of sides (or vertices):

## How To Find The Diagonal of a Rectangle

Test this formula with something we know: diagonals of a rectangle. A rectangle has four sides and four vertices.

Be skeptical! Try it for a pentagon (five sides):

A pentagon has only five diagonals; our formula works.

Be *really* skeptical! Try it for a tetracontakaiheptagon, which is a ridiculously long (but correct) name for a 47-gon:

Trust the formula. A 47-gon has 1,034 diagonals. This formula works every time to tell you exactly how many diagonals can be constructed inside (or outside) of any simple polygon, whether the shape is convex or concave.

## Diagonal of a Rectangle Formula

Where a is the side of the square.

### Diagonal of a Cube Formula

For a cube, we find the diagonal by using a three-dimensional version of the Pythagorean Theorem/distance formula:

## Lesson Summary

You have learned a lot about particularly important parts of polygons, their diagonals. You now know how to identify the diagonals of any polygon, what some real-life examples of diagonals are, and how to use the formula,

## What is Diagonal?

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:

(image will be uploaded soon)

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.

**The Given Table Shows the Number of Diagonals in Different Polygons**

Let us study diagonal formula for different shapes

**Diagonal of Square Formula**

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 =d_{1} and y = d_{2} are the diagonals of a parallelogram and a and b are the two sides.

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

**Diagonal of a Cube Formula**

For a cube, we find the diagonal by using a three-dimensional version of the Pythagorean Theorem/distance formula:

Where l, b, and h are length, breadth, and height of a cuboid

**Solved Examples**

**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 = 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,

## FAQs (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.

**Example 1:**

*Find the total number of diagonals contained in an 11-sided regular polygon.*

**Solution:**

In an 11-sided polygon, total vertices are 11. Now, the 11 vertices can be joined with each other by 11C2 ways i.e. 55 ways.

Now, there are 55 diagonals possible for an 11-sided polygon which includes its sides also. So, subtracting the sides will give the total diagonals contained by the polygon.

So, total diagonals contained within an 11-sided polygon = 55 -11 i.e. 44.

**Formula Method:**

According to the formula, number of diagonals = n (n-3)/ 2.

So, 11-sided polygon will contain 11(11-3)/2 = 44 diagonals.

**Example 2:**

*In a 20-sided polygon, one **vertex does not send any diagonals. Find out how many diagonals does that 20-sided polygon contain.*

**Solution:**

In a 20-sided polygon, the total diagonals are = 20(20-3)/2 = 170.

But, since one vertex does not send any diagonals, the diagonals by that vertex needs to be subtracted from the total number of diagonals.

In a polygon, it is known that each vertex makes (n-3) diagonals. In this polygon, each vertex makes (20-3) = 17 diagonals.

Now, since 1 vertex does not send any diagonal, the total diagonal in this polygon will be (170-17) = 153 diagonals.

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