Area of Rhombus & its Proof | Mensuration | Concept Clarification | Very Important Concept |

Описание: Area of Rhombus & Its Proof | Mensuration | Concept Clarification | Very Important Concept |

One must watch this lecture. This method explained in the video is very helpful for making maths calculation easy specially in competitive exams.

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What is the Area of a Rhombus?
The area of a rhombus can be defined as the amount of space enclosed by a rhombus in a two-dimensional space. To recall, a rhombus is a type of quadrilateral projected on a two dimensional (2D) plane, having four sides that are equal in length and are congruent. It is also known as equilateral quadrilateral because all its four sides are equal in nature.

Area of a Rhombus Formula Derivation
Area of the rhombus is the sum of areas of 2 triangles

Consider the Rhombus ABCD.

Let E be the point of intersection of two diagonals.

The area of the rhombus ABCD = Area of ∆ ADC + Area of ∆ ABC

Area of the rhombus = 2 × Area of ∆ ABC ---(1) (∵ ∆ ABC congruent to ∆ ADC)

Area of ∆ ABC
=
1
2

×Base×Height =
1
2

×AC×BE =
1
2

×AC×
1
2

×BD(∵BE=
BD
2

) =
1
4

(AC×BD)−−−(2)
Area of the Rhombus ABCD
=2×
1
4

×AC×BD =
1
2

×AC×BD(From (1) and (2)) =
1
2

×diagonal 1×diagonal 2
Area of a Rhombus =
1
2

× diagonal 1 × diagonal 2 units 2


Area of a Rhombus with Sides given
Rhombus is a parallelogram.

We know that, the area of a parallellogram =base×height sq units.

The same is applied to rhombus as well.

Area of a Rhombus = base × height sq units
Let the side be 7 inches and height of the rhombus be given as 10 inches.

Let us find its area.

Area =base×height units2 =7×10 inches2 =70 inches2
Area of a Rhombus When Sides and Angles Are Given
area of the rhombus is the product of the square of the sides and the sine of an angle.

Area of a Rhombus = side 2×sin(A)sq. units
where, a is any interior angle.

We can use any angle because either the angles are equal or they are supplementary, and supplementary angles have the same sine.

If the length of its side is 4 yards and one of its angle A is 30 ∘, then,

Area =s2×sin(30) =42×
1
2

=16×
1
2

Area =8 sq yards
How to Calculate the Area of a Rhombus
Explanation
Consider the rhombus ABCD.

AB, BC, CD, DA are the congruent (equal) sides.

AC and BD are the diagonals, and they meet at E.

Given CD= 17 feet and AE = 8 feet.

area of the rhombus given its diagonals

Now we know diagonal 1, AC = 16 feet.

Next, we need to calculate BD.

BD=BE+ED=2×BE
We still have an unknown, BE.

Pythagorean Theorem states that

BC2=BE2+EC2
.

BC = 17 feet (∵ CD =BC as all the sides are congruent)

EC = 8 feet (∵ AE = EC as the diagonals bisect each other)

172 =BE2+82 BE2 =289−64 =225 BE =15 feet BD =30 feet
It's time to substitute all the values in the area of rhombus calculator formula.

Area of the Rhombus
=
1
2

×d 1×d 2 sq units =
1
2

×BD×AC sq feet =
1
2

×16×30 sq feet Area =240 sq feet
Solved Examples
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Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Q1. John has too many formulae in his mind and is confused which is the one for the area of a rhombus. Can you help him find the incorrect one in the list?

A = d1xd2/2
A = d1xd2/2

A = 1/2 x (sum of parallel sides) x height
A = 1/2 x (sum of parallel sides) x height

A= base x height
A= base x height

A= base x base x Sine of an angle


Q 2. Henry comes around the ground which is rhombus in shape to fix the poles. He has taken 48 feet to come around and he measures the height of the ground as 8 feet. Help him find the area of the ground.

192 sq. ft
192 sq. ft

96 sq. ft
96 sq. ft

32 sq.ft
32 sq.ft

384 sq. ft







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