how to find the area of a rhombus

Area of Rhombus embibe.com

Area of Rhombus: The rhombus is a quadrilateral with a shape like a diamond. It is a quadrilateral that has two sets of parallel sides, equal opposite sides, and equal opposite angles. It is also called an equilateral quadrilateral because all its four sides are equal.

Rhombus is observed in several things around us, such as a kite, windows of a car, rhombus-shaped earrings, buildings structure, diamonds, mirrors, and even a section of the baseball field. The properties of rhombus and its area formula are widely used to solve various problems. Let's discuss them one by one.

What is a Rhombus?

The word rhombus is derived from the Greek word "rhombos" which means a piece of wood whirled on a string to create a roaring noise, and this word was eventually derived from the Greek verb "rhembo" which means to turn round and round.

The rhombus is an equal opportunity parallelogram. No side is bigger or smaller than the other. All sides are equal, and opposite sides are parallel. The rhombus is the subset of the parallelogram. The difference between a square and a rhombus is that all angles of a square are right angles, but the angles of a rhombus need not be right angles. So, a rhombus with right angles becomes a square. Hence, every square is a rhombus, but all rhombuses are not a square.

Properties of a Rhombus

1. All sides of a rhombus are equal i.e., \(AB = BC = DC = DA.\)
2. Diagonals bisect each other at \(90^\circ .\)i.e., diagonals \(BD\) and \(AC\) bisect each other at \(90^\circ .\)
3. Opposite sides are parallel in a rhombus i.e., \(AB\parallel CD\) and \(AD\parallel BC.\)
4. Opposite angles are equal in a rhombus i.e., \(\angle A = \angle C,\angle B = \angle D.\)
5. Adjacent angles add up to \(180^\circ ,\) i.e., \(\angle A + \angle B\; = 180^\circ ,\angle B + \angle C\; = 180^\circ ,\angle C + \;\angle D = 180^\circ ,\;\angle A + \angle D\; = 180^\circ .\)

What is the Area of Rhombus?

The area of a rhombus can be interpreted as the amount of space enclosed by a rhombus in a \(2D\) space. The area of the rhombus can be calculated in three different ways.

Area of a rhombus with diagonals
Area of a rhombus with side
Area of rhombus using trigonometry concept

Area of Rhombus Formula

Area of rhombus has different formulas in different cases, and the most commonly used are given below:

Area of rhombus using diagonals	Area \( = \frac{1}{2} \times {d_1} \times {d_2}\)
Area of rhombus using base and height	Area \( = b \times h\)
Area of rhombus using trigonometry	Area \( = {b^2} \times \sin \left( a \right)\)

Where,
1. \({d_1} = \) Length of diagonal \(1.\)
2. \({d_2} = \) Length of diagonal \(2.\)
3. \(b = \) Length of any side
4. \(h = \)Height of rhombus
5. \(a = \) Measure of any interior angle

Area of Rhombus with Diagonals

Half of the product of diagonals provides us with the area of the rhombus. Area \(= \frac{1}{2} \times {d_1} \times {d_2}\)
Where
\({d_1} = \)Length of diagonal \(1.\)
\({d_2} = \) Length of diagonal \(2.\)

Let \(O\) be the point of intersection of diagonals \({d_1}\) and \({d_2}.\)
The diagonals bisect each other at \({90^ \circ },\) i.e., diagonals, \(AC\) and \(BD\) bisect each other at \({90^ \circ }.\)
So, \(OA = OB = OC = OD\) and \(AOB = BOC = COD = DOA.\)
So, the area of \(AOB = \) area of \(BOC = \) area of \(COD = \) area of \(DOA.\)
Hence, area of rhombus \(ABCD = \) area of \(AOB + \) area of \(BOC + \) area of \(COD + \) area of \(DOA.\) So, the area of rhombus \(ABCD = 4\) times the area of \(\Delta AOB\)
\(= 4 \times \frac{1}{2} \times OA \times OB\)
\( = 4 \times \frac{1}{2} \times \frac{{{d_1}}}{2} \times \frac{{{d_2}}}{2}\)
\( = \frac{1}{2} \times {d_1} \times {d_2}\)

Area of Rhombus With Side (When Its Base and Height are known)

When the length of base (side) and height of the rhombus are given, the area of a rhombus with a side is easy to find. A simple formula can be applied, and the area of a rhombus with a side is calculated. If the base (side) of the rhombus is named \(b\), height as \(h\), then the product of base and height constitutes the area of the rhombus.
Area of Rhombus \( = {\text{Base}} \times {\text{Height}}\)
\( = b \times h\)

Area of Rhombus Using Trigonometry Concept

In some cases where height is unknown, but the base and one of the angles is known, the area can be calculated by multiplying the square of the base with the sine of that angle. If the base of the rhombus is \(b\) and the measure of interior angle \(A\) is \(a\), then
Area of rhombus \(ABCD = {b^2} \times \sin \left( a \right)\)

Let \(O\) be the point of intersection of two diagonals.
The diagonals bisect each other at \({90^ \circ },\) i.e., diagonals, \(AC\) and \(BD\) bisect each other at \({90^ \circ }.\)
So, the area of rhombus \(ABCD = \) area of \(\Delta AOB + \) area of \(\Delta BOC + \) area of \(\Delta COD + \) area of \(\Delta DOA\)
So, the area of Rhombus \(ABCD = 2 \times \) area of \(\Delta BAD\)
\(= 2 \times \frac{1}{2} \times {b^2} \times \sin \left( a \right)\)
\(= {b^2} \times \sin \left( a \right)\)

Area of Rhombus in Vector form

The vector concept is also used to calculate the area of the rhombus. Since all rhombuses are parallelograms, the area of the rhombus in vector form is given by
Area of rhombus \(= \left| {\overrightarrow a \times \overrightarrow b } \right|\) where \({\overrightarrow a }\) and \({\overrightarrow b }\) are any two adjacent sides of a rhombus.

Solved Examples – Area of Rhombus

Question 1: Evaluate the area of a rhombus, if its base is \(8\,{\rm{cm}}\) and height is \({\rm{5}}\,{\rm{cm}}{\rm{.}}\)
Answer:
Given,
Base, \(b = 8\,{\rm{cm}}\)
Height, \(h = 5\,{\rm{cm}}\)
Area, \(A = b \times h\)
\( = 8 \times 5\,{\rm{c}}{{\rm{m}}^2}\)
\( = 40\,{\rm{c}}{{\rm{m}}^2}\)

Question 2: Evaluate the area of the rhombus having diagonals \(5\,{\rm{cm}}\) and \(7\,{\rm{cm}}{\rm{.}}\)
Answer:
Given that,
Diagonal \({d_1} = 5\,{\text{cm}}\)
Diagonal \({d_2} = 7\,{\text{cm}}\)
Area of rhombus \( = \frac{1}{2} \times {d_1} \times {d_2}\,{\rm{c}}{{\rm{m}}^2}\)
\( = \frac{1}{2} \times 5 \times 7\,{\rm{c}}{{\rm{m}}^2}\)
\( = \frac{{35}}{2}{\rm{c}}{{\rm{m}}^2}\)
\( = 17.5\,{\rm{c}}{{\rm{m}}^2}\)

Question 3: Find out the area of a rhombus if the length of its side is \({\rm{4}}\,{\rm{cm}}\) and one of its angle \(A\) is \({30^ \circ }\).
Answer:
Given,
Side or base \(= b = 4\,{\text{cm}}\)
\(A = {30^ \circ }\)
Area of rhombus \( = {b^2} \times \sin \left( {{{30}^ \circ }} \right){\rm{c}}{{\rm{m}}^2}\)
\( = 16 \times 0.5\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)
\( = 8\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

Question 4: Calculate the area of the rhombus having side equal to \({\rm{17}}\,{\rm{cm}}\) and one of its diagonals is \({\rm{16}}\,{\rm{cm}}{\rm{.}}\)
Answer:
\(ABCD\) is a rhombus,
So , \(AB = BC = CD = DA = 17\,{\text{cm}}\)
\(AC = 16\,{\text{cm}}\)
Let \(O\) be the intersection point of diagonals.
So, \(AO = 8\,{\text{cm}}\)
In \(\Delta DOA\)
\(A{D^2} = A{O^2} + O{D^2}\)
\( \Rightarrow {17^2} = {8^2} + O{D^2}\)
\( \Rightarrow 289 = 64 + O{D^2}\)
\( \Rightarrow 225 = O{D^2}\)
\( \Rightarrow OD = 15\)
Therefore, \(BD = 2 \times OD\)
\( = 2 \times 15\)
\( = 30\,{\rm{cm}}\)
Now, area of rhombus \(= \frac{1}{2} \times {d_1} \times {d_2}\)
\( = \frac{1}{2} \times 16 \times 30\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)
\({\rm{ = 240}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

Question 5: The building floor consists of \({\rm{1000}}\) tiles, which are rhombus in shape, and each of its diagonals is \({\rm{80}}\,{\rm{cm}}\) and \({\rm{50}}\,{\rm{cm}}\) in length. Calculate the total cost of polishing the floor if the cost per \({{\rm{m}}^2}\) is \(₹5\).
Answer:
In each rhombus-shaped tile, the length of the diagonals is \({\rm{80}}\,{\rm{cm}}\) and \({\rm{50}}\,{\rm{cm}}{\rm{.}}\)
Therefore, the area of each tile \( = \frac{1}{2} \times 80 \times 50 = 2000\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)
Therefore, the area of \(1000\) tiles \({\rm{ = 1000 \times 2000}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)
\({\rm{ = 2000000}}\,{\rm{c}}{{\rm{m}}^2}\)
\({\rm{ = 200}}\,{{\rm{m}}^2}\)
For \({\rm{1}}\,{{\rm{m}}^{\rm{2}}}\) cost of polishing \(= ₹5\)
For \(200\,{{\rm{m}}^{\rm{2}}}\) cost of polishing \(= ₹5 \times 200\)
\( =₹1000\)

Question 6: If the area of a rhombus is \({\rm{196}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\) and the length of one of the diagonals is \({\rm{24}}\,{\rm{cm}}\) find the length of the other diagonal.
Answer:
Area of rhombus \({\rm{ = 192}}\,{\rm{c}}{{\rm{m}}^2}\)
That is, \(\frac{1}{2} \times {d_1} \times {d_2} = 192\)
On putting \(24\) for \({d_1},\) we get
\( \Rightarrow \frac{1}{2} \times 24 \times {d_2} = 192\)
\( \Rightarrow 12 \times {d_2} = 192\)
\( \Rightarrow {d_2} = 16\)
So, the length of the other diagonal is \({\rm{16}}\,{\rm{cm}}{\rm{.}}\)

Summary

A rhombus is a quadrilateral whose all sides are equal and opposite sides are parallel. Rhombus is a subset of a parallelogram and it can be square in a special case. The properties of the rhombus and the area of the rhombus formula are widely used to solve real-life problems. Every formula is derived from basic geometrical concepts.

Frequently Asked Questions (FAQs)

Q.1. What is the formula for finding the area of a rhombus?
Ans: The most basic formula used to calculate the area of a rhombus is
\( = \frac{1}{2} \times {d_1} \times {d_2}\)
where \({d_1} = \) Length of diagonal \(1,\,{d_2} = \) Length of diagonal \(2.\)

Q .2. What is the formula for the perimeter and area of a rhombus?
Ans: The perimeter of a rhombus \( = 4 \times \) Side length of a rhombus. Since all sides of the rhombus are equal, its perimeter formula is the same as that of the square.
The most basic formula used to calculate the area of a rhombus
\( = \frac{1}{2} \times {d_1} \times {d_2}\)
where \({d_1} = \) Length of diagonal \(1,\,{d_2} = \)Length of diagonal \(2\).

Q.3. What are the \(4\) properties of a rhombus?
Ans: The most commonly \(4\) properties of a rhombus are given below:
1. All sides of a rhombus are equal.
2. Diagonals bisect each other at \({90^ \circ }\).
3. Opposite sides are parallel, and opposite angles are equal in a rhombus Adjacent angles add up to \({180^ \circ }\).

Q.4. What is the difference between rhombus and square?
Ans: Although a rhombus and a square have all their sides equal, they are not the same. All interior angles of the square are \({90^ \circ }\) but that is not necessary in the case of the rhombus. In the rhombus, adjacent angles add up to \({180^ \circ }\). Every square is a rhombus, but all rhombuses are not squares.

Q.5. What does a rhombus look like?
Ans: A rhombus looks like a diamond, symmetrical kite, Kaju Katli, etc.

Q.6. How do you find the area of a rhombus with one side and one diagonal?
Ans: The diagonals of a rhombus bisect each other at \({90^ \circ }\). So from the given side and a given diagonal, we can find the other diagonal by applying the Pythagoras theorem. When diagonals are obtained, we can use the formula of area of rhombus \(= \frac{1}{2} \times {d_1} \times {d_2}\).

Q.7. Are the diagonals of the rhombus equal?
Ans: Generally, the diagonals of the rhombus are not equal, but if the rhombus is a square, its diagonals are equal. So in a particular case, it can be equal.

Q.8. Are all rhombuses parallelograms?
Ans: Yes, opposite sides are parallel and equal in a parallelogram. So, all rhombuses are parallelograms as all sides of a rhombus are equal, and opposite sides are parallel.

Q.9. How to find the area of a rhombus if the base and height are given?
Ans: Area of a rhombus (if base and height are known) \(= \) base of the rhombus \( \times \) height of the rhombus.

Q.10. Is every square a rhombus?
Ans: Yes, every square is a rhombus as the rhombus is a quadrilateral having all its sides are equal and opposite sides are parallel. So a square is a special case of rhombus where all its angles are also \({90^ \circ }\).

Q.11. Do all angles of a rhombus add up to \({360^ \circ }\)?
Ans: Yes, since opposite interior angles of a rhombus are supplementary. The four interior angles of any rhombus always add up to \({360^ \circ }\).

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