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Diagonal Formula

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Diagonal Formula is the formula that is used to calculate the length of the diagonal of any polygon. A diagonal is a slant line that joints two alternate sides of any polygon and finding its length is very important for solving various problems of geometry.

There are different ways for finding the diagonal of any polygon and we use various techniques for that purpose. Using Pythagoras Theorem is one such way. We have to study the Diagonal Formula in Class 8. In this article, we will learn about the Diagonal definition, its formula, various examples, and others related to it.

What is Diagonal?

Diagonal is defined as the line segment formed by joining two vertices of the polygon that are not on the same edge. The word is derived from the word “Diagnosis” which has a Greek origin and means “From Angle to Angle”.

There are various properties of the diagonals that are added below.

Properties of Diagonals

Various properties of the Diagonal of any polygon are

  • Diagonals of a regular polygon are congruent.
  • Diagonals of any polygon are not on the same edge, etc.

Diagonal Formula Definition

Diagonal Formula is the formula that is used to find the length of the diagonal of any polygon. There is no one particular formula used to find the length of the diagonals we use different methods to find the length of the diagonal of the polygon.

The number of diagonals that any n-sided polygon has is given using the formula,

Number of Diagonals = n(n-3)/2

This is also called the diagonal formula.

Diagonal Formula For Square

The diagonal of the Square is the line formed by joining two opposite vertices of the square. The formula for finding the length of the diagonal of the square is,

Square Diagonal: a√(2)

where,

  • a is the Length of Side of Square

The diagonal of a square is shown in the image added below,

Diagonal-of-a-Square

Here, suppose the length of side AB = BC be “a units” then the length of diagonal AC is “a√(2) units”.

Diagonal Formula For Rectangle

The diagonal of the Rectangle is the line formed by joining two opposite vertices of the rectangle. The formula for finding the length of the diagonal of the rectangle is,

Rectangle Diagonal: √[l2 + b2]

where,

  • l is the Length of Rectangle
  • b is the Breadth of Rectangle

The diagonal of a rectangle is shown in the image added below,

Diagonal-of-a-Rectangle

Here, suppose the length of side AB = l units, BC = b units then the length of diagonal AC is “√[l2 + b2] units”.

Diagonal Formula of Rhombus

The diagonal of the Rhombus is the line formed by joining two opposite vertices of the rhombus. The formula for finding the length of the diagonal of the rhombus is,

Rhombus Diagonal = 2(A)/q

Rhombus Diagonal = 2(A)/p

where,

  • p and q are Diagonals of Rhombus
  • A is Area of Rhombus

The diagonal of a rhombus is shown in the image added below,

Diagonal-of-a-Rhombus

Here, suppose the area of the rhombus is A units2 and its diagonal BD is ‘p’ units then length of its other diagonal AC is, 2(A)/p.

Diagonal Formula of Parallelogram

The diagonal of the Parallelogram is the line formed by joining two opposite vertices of the parallelogram. The formula for finding the length of the diagonal of the parallelogram is,

p = √(x2 + y2 – 2xycos A) = √(x2 + y2 + 2xycos B)

q = √(x2 + y2 + 2xycos A) = √(x2 + y2 – 2xycos B)

where,

  • p2 + q2=2(a2 + b2)
  • x and y are Sides of Parallelogram
  • p and q are Two Diagonals of Parallelogram

Read More,

Examples of Diagonals Formula

Example 1: Find the length of the diagonals of a square whose side measures 10cm.

Solution:

Given,

Length of Side of Sqaure = 10 cm

Diagonal of Square = a√(2)

=> 10√(2)

=> 14.14 cm

Thus, the length of diagonal of square is 14.14 cm.

Example 2: Find the second diagonal of a rhombus having its first diagonal measuring 20 cm and the area of the rhombus is 200 cm2 

Solution:

Given,

Let two diagonals be p and q

Length of diagonal p = 20 cm

Area of Rhombus = 200 cm2

q = 2(A)/p

=> q = 2 × 200/20

=> q = 400/20

=> q =  20 cm

Thus, the other diagonal of the Rhombus is 20 cm.

Example 3: Find the first diagonal of a rhombus having its second diagonal measuring 5cm and the area of the rhombus is 100cm2.

Solution:

Given,

Let two diagonals be p and q

Length of One diagonal q = 5 cm

Area of Rhombus = 100 cm2

p = 2(A)/q

=> p = 2 × 100/5

=> p = 200/5

=> p = 40 cm

Thus, the other diagonal of the Rhombus is 40 cm.

Example 4: Find out the diagonal length of a rectangle having its length 20cm and breadth of 12cm.

Solution:

Given,

  • Length of Rectangle = 10 cm
  • Breadth of Rectangle = 3 cm

We have,

Diagonal of Rectangle = √(l2 + b2)

=> √(20)2 + (12)2

=> √(400+144)

=> √(544)

=> 23.32 cm

The length of diagonal of recatngle is 23.32 cm

Practice Questions on Diagonal Formula

Q1: Find the length of diagonal of Rectangle whose length is 12 cm and breadth is 5 cm.

Q2: Find the length of diagonal of Square whose each side measures 6 cm.

Q3: Find the length of side of a Square the length of whose Diagonal is 10 cm.

Q4: Find the length of the diagonal of Rhombus whose one diagonal is 3 cm, and the side is 4 cm.

FAQs on Diagonals Formula

1. What is Diagonal of a Polygon?

Diagonal of a Polygon is defined as the line segment joining two vertices of the polygon that are not on the same edge. A polygon with n sides have n(n-3)/2 diagonals, i.e. for square we have 2 diagonals, for pentagon we have 3 diagonals, etc. We have to note that a triangle has no diagonals.

2. What is Formula for Finding Number of Diagonals of a Polygon?

The formula for finding the number of diagonals for any polygon is n(n-3)/2 where,

n represent the number of Diagonals of the Polygon.

3. What is the Formula for Finding Length Diagonals?

A quadrilateral with n sides has n(n-3)/2 number of diagonals and finding the length of the diagonals required using different formulas as there is no one single formula to find the length of diagonal of a polygon. Different formula for finding length of diagonal of polygons are,

  • Length of Diagonal of Sqaure = a√(2) where, a is the length of side of rectangle.
  • Length of Diagonal of Pectangle = √(l2 + b2) where, l and b are length and breadth of Recatangle respectively.

Apart form them other formulas are also used for finding length of diagonal of polygon.

4. What is Formula for Diagonal of Square?

Square is a quadrilateral in which all four sides and all four angles are equal and opposite sides are parallel. Suppose we have a square with side “a” then, the formula for finding the length of diagonal of the square is a√(2) units.

5. What is Formula for Diagonal of Rectangle?

Rectangle is a quadrilateral in which all four angles are equal and opposite sides are equal and parallel. Suppose we have a rectangle with length “l” and breadth “b” then, the formula for finding the length of diagonal of the rectangle is √(l2 + b2) units

6. What is the Diagonal Formula of Cuboid?

Cuboid is a 3D solid figure whose length breadth and height are unequal. The length of diagonal of cuboid is given as d = √(l2 + b2 + h2) where l, b and h are length, breadth and height of the cuboif respectively

7. What is Diagonal Formula of Cube?

Cube is 3D solid figure whose all dimensions are equal. The formula for length of body diagonal of a cube is given as √3⨯side and the formula for length of face diagonal of a cube is given as √2⨯side



Last Updated : 10 Jan, 2024
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