Area of Parallelogram: Properties & Formula

Area of a parallelogram: It is significantly important for students to learn the basic formulas of area and perimeter. The area of a parallelogram is referred to as the space covered by a parallelogram in a two-dimensional plane. Any four-sided shape will have two pairs of opposite “parallel” sides. A parallelogram is a kind of quadrilateral. A parallelogram is formed when a quadrilateral has two parallel opposing sides.

Suppose you built a carton to hold, say, clothes, and you forget to put a bottom on it. Two of the carton’s bottom opposite sides are \(10\) inches, and the other two are \(15\) inches. If you turn the carton so one of its \(10\)-inch sides is flat on a table, the carton will naturally change its shape (because it had no bottom to hold the four sides rigid). The carton’s bottom then changes to a parallelogram. If you push or pull the carton, every shape it takes is a parallelogram.

Area of Parallelogram: Definition

As a quick refresher, the area of a parallelogram is the area covered by a parallelogram in a \(2D\) planar region. A parallelogram is a special type of quadrilateral. The opposite sides of a parallelogram are equal and parallel. Now if you are wondering what a quadrilateral is, we will jump into that part a little later.

Area of a Parallelogram Calculator

Area of a parallelogram is the product of base and height.
\(A = b \times h\)

Isn’t the above formula to find the area of a rectangle? Are they both the same? Seems confusing, right?

Let us understand with the help of the diagram given below.

We transformed a parallelogram into a rectangle with the same base and height. Now, since the bases and heights of the parallelogram and the rectangle are the same:

Area of a parallelogram \(= \) Area of a rectangle\(AEFD\)
\(= EF \times AE = BC \times AE\)
\(=\mathrm{base}\times\mathrm{height}\)

To calculate the area of a parallelogram, multiply the perpendicular’s base by its height. We can see that the parallelogram’s base and height are perpendicular to one another, but the lateral side is not perpendicular to the base. 

Is this the only way to find the area of parallelogram?

The area of any parallelogram can also be calculated using its diagonal lengths.

Area of a Parallelogram Using Diagonals

We know that there are two diagonals of a parallelogram, which intersect each other. Suppose the diagonals intersect each other at an angle. In this case, the area of the parallelogram is given by:

Area of a parallelogram with diagonals \(\ ” {p ”}\) and \(\ ” {q ”}\)\(= \frac12 \times p \times q\sin (x)\)

Are we done here with the area of parallelogram?

No, there is another formula to evaluate the area of a parallelogram when height is not given.

Area of a Parallelogram Without Height

In case, if the height of the parallelogram is not given, then we can make use of the trigonometric concept to find its area.

Area of a parallelogram \(= ab\sin \theta ,\) where \(a\) and \(b\) are the length of the parallel sides and θ is the angle between the sides of the parallelogram.

There is one more way to find the area of a parallelogram or more precisely we can say to find the area of the irregular quadrilaterals.

1. Determine all the sides of irregular shapes. Make sure all the sides should be in the same unit.
2. Make use of the appropriate measuring tool and draw the area on a piece of paper using the measurements you obtained.
3. Divide the drawing into different shapes like squares or rectangles etc.
4. Calculate the area of each shape by using their respective formulas.
5. Finally, add the areas of all the individual shapes to find the area of the irregular quadrilateral.

As discussed earlier and as the name suggests, a parallelogram is a quadrilateral formed by two pairs of parallel lines with equal measures. Thus, it is a four-sided figure. The sum of interior angles in a parallelogram is \({360^ \circ }\). Since a parallelogram is a \(\left( {2 – D} \right)\) figure, it has an area and perimeter.

Parallelogram is a Quadrilateral

Yes, a parallelogram is a special type of a quadrilateral.
The following properties make it unique:
1. Opposite sides are equal.
2. Opposite angles are equal.
3. Adjacent angles are supplementary \((\)i.e., equal to \({180^ \circ }\))
4. If one of the angles is right angle, then all the other angles are also right angles.
5. The diagonals of a parallelogram bisect each other.
6. Each diagonal separates it into two congruent triangles.

Types of Quadrilateral

A closed plane figure with \(4\) straight sides is called a quadrilateral. The above diagram shows some quadrilaterals found in our daily lives. In this section, we will learn about the properties of some special quadrilaterals. For instance, rectangles, squares, rhombuses, and trapeziums.

Also, we will check whether these quadrilaterals are parallelograms or not with the help of their properties.

Let us recall their definitions.

(A) Rhombus

A rhombus is a parallelogram with four equal sides.

Try Recalling the properties of a parallelogram and relate with properties of rhombus mentioned below:
1 All sides of rhombus are equal. Hence, opposite sides are equal.
2 The opposite sides of a rhombus are parallel.
3 The sum of two adjacent angles is equal to \({\rm{18}}{{\rm{0}}^ \circ }.\)
It possesses the properties of a parallelogram.
Hence, a rhombus is a parallelogram.

(B) Trapezium

1 A trapezium has two parallel sides (mandatory) and two non-parallel sides (optional).
2 The sum of all the \(4\) angles is equal to \({360^ \circ }.\)
3 Two pairs of adjacent angles of a trapezium formed between the parallel sides and one of the non-parallel sides is equal to \({\rm{18}}{{\rm{0}}^ \circ }.\)
So, all trapezium cannot be a parallelogram.

Now, if we read the first property of a trapezium, which is, trapezium has one pair of opposite sides parallel. And, thus, it makes trapezium stand out of the queue of not being a parallelogram because a parallelogram has two pairs of parallel sides.

So, all trapezium cannot be a parallelogram.

(C) Kite

A kite is a quadrilateral whose \(4\) sides can be grouped into \(2\) pairs of equal sides that are adjacent to each other.

1 The two angles are equal where the unequal sides meet.
2 It has \(2\) diagonals that intersect each other at \({90^o}.\)
3 A kite is symmetrical about its main diagonal.

Now, here is the twist.

Read the definition of kite again. A kite has \(2\) pairs of equal sides that are adjacent to each other. In contrast, a parallelogram too has \(2\) pairs of equal sides, but instead of being adjacent they are opposite to each other.

So, a kite quadrilateral cannot be a parallelogram.

Parallelogram Diagram

Unit of Area of a Parallelogram

The area of a parallelogram is the amount of space it occupies or encloses in the plane. The area is usually measured in square units like square metres, square feet, square inches, etc.

This article discussed that a parallelogram is a special type of quadrilateral with two pairs of parallel sides. The opposite angles are of equal measure, and opposite sides are of equal length, and since a parallelogram is a ({2-D}) figure, it has an area and perimeter. Also, this article gives the various ways to find the area of a parallelogram and different formulas, depending on the given data. In the below section, we have also provided solved examples and FAQs on the area of the parallelogram, which will be helpful to you to understand the concept better.

Solved Examples on Area of Parallelogram

Example 1: Find the area of a parallelogram with base (10 cm) and height (6 cm)
Solution: \( {A=B \times H}\)
\(A = \left( {10\,{\rm{cm}}} \right) \times \left( {6\,{\rm{cm}}} \right)\)
\(A = 60\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

Example 2: The area of a parallelogram is \(24\) square centimeters and the base is \(4\, {\text{cm }}\). Find the height.
Solution: \(A = B \times H\)
\(24\,{\text{c}}{{\text{m}}^2} = \left({4\,{\text{cm}}}\right) \times H\)
\(24\,{\text{c}} { {\text{m}}^2} \div \left ({4\, {\text{cm}}} \right) = H\)
\(H = 6\, {\text{cm}}\)

Example 3: Find the base of a parallelogram if its area is \(60\) square cm and its altitude is \(15\,{\text{cm}}\)
Solution: Area of a parallelogram \( {\text{=60}}\, {\text{c}} {{\text{m}}^2}\)
\( {\text{b}} \times {\text{h=60}}\,\)
Here, altitude (or) height \( {\text{h=15}}\, {\text{cm}}\)
\(\mathrm{Area}=\mathrm{base}\times\mathrm{height}\)
\( {\text{b=4}}\,{\text{cm}}\)
So, the base of the parallelogram is \( {\text{4}}\,{\text{cm}}\)

Example 4: The base of the parallelogram is thrice its height. If the area is \( {\text{147}}\, {\text{c}}{ {\text{m}}^2}\) Find the base and height.
Solution: Let the height of the parallelogram \(= x\, {\text{cm}}\)
Then the base of the parallelogram \(= 3x\,{\text{cm}}\)
Area of the parallelogram \(= 147\, {\text{c}}{{\text{m}}^2}\)
Area of parallelogram \( = \mathrm{base}\times\mathrm{height}\)
\(147 = 3x \times x\)
\(\Rightarrow 3{x^2} = 147\)
\(\Rightarrow {x^2} = 49\)
\(\Rightarrow x = 7\)
Therefore, \(3x = 3 \times 7 = 21\)
Therefore, base of the parallelogram is \(21\,{\text{cm}}\) and height is \(7\,{\text{cm}}\)

Example 5: The adjacent sides of a parallelogram are \(5\,{\text{m}}\) and \(4\,{\text{m}}\) If the distance between the longer sides is \(2\, {\text{m}}\), find the distance between the shorter sides.
Solution: Let \(ABCD\) be a parallelogram with side \({\text{DC=5}}\,{\text{m}}\) and corresponding altitude \(AE = 2\,{\text{m}}\)

The adjacent side \(AD = 4m\) and the corresponding altitude is \({CF}{.}\)
We know that area of a parallelogram \(= Base \times Height\)
We have two altitudes and two corresponding bases.
So, equating them we get
\({AD} \times {CF} = {DC} \times {AE}\)
\(4 \times CF = 5 \times 2\)
\(CF = \left( {\frac{{5 \times 2}}{4}} \right) = 2.5{\mkern 1mu} \;{\rm{m}}\)
Hence, the distance between the shorter sides is \(2.5\, {\text{m}}\)

Example 6: A floral design on the floor of a building consists of \(280\) tiles. Each tile is in the shape of a parallelogram of altitude \(6\,{\text{cm}}\) and base \(10\,{\text{cm}}\) Find the cost of polishing the design at the rate of \(₹= 0.5\,{\text{per}}\,{\text{c}}{m^2}\)
Solution: Given altitude of a tile \( = 6cm\)
Base of a tile \( = 10m\)
Area of one tile \(= Altitude \times Base \)
\({\rm{ = 6}}\,{\rm{m \times 10}}\,{\rm{m = 60}}\,{{\rm{m}}^{\rm{2}}}\)
Area of \(280\) tiles \({\rm{ = 280 \times 60}}\,{{\rm{m}}^{\rm{2}}}{\rm{ = 16800}}\,{{\rm{m}}^{\rm{2}}}\)
Rate of polishing the tile \( {\rm{ = ₹0}}{\rm{.5}}\,{\rm{per}}\,{{\rm{m}}^2}\)
Thus, total cost of polishing the design \(= 16800 \times 0.5 = ₹8400\)

Example 7: Find the area of a parallelogram if its two parallel sides are \(80\,{\text{m}}\) and \(40\,{\text{m}}\) and the angle between them is \( {30^ \circ }\)
Solution: Let \(a=80\,{\text{m}}\) and \(b=40\,{\text{m}}\)
The angle between \(a\) and \(b,\theta = {30^\circ }\)
Area \(= ab\sin \theta \)
After substituting the values, we get
\(A = 80 \times 40 ({\sin \ 30^\circ)}\)
\(A = 3200 \times \frac{1}{2}\)
\(A=1600\,{\text{cm}}\)

Summary

This article discussed a parallelogram, its shape, formula to find the area of a parallelogram with solved examples. The opposite angles are of equal measure, opposite sides are of equal length, and since a parallelogram is a (2-D) figure, it has an area and perimeter.
Also, this article gives the various ways to find the area of a parallelogram and different formulae, depending on the given data. Also, we discussed which other quadrilaterals have the same properties as a parallelogram and fall under the same category.

FAQs

Q.1. What is a parallelogram?
Ans: A parallelogram is a four-sided geometrical shape made by two parallel lines. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure.

Q.2. Are the diagonals of a parallelogram equal?
Ans: Yes, parallelograms have equal diagonals.

Q.3. What is the area of parallelogram formula?
Ans: A parallelogram is a quadrilateral formed by two pairs of parallel lines with equal measures. The area of a parallelogram is the area covered by a parallelogram in a \(2 – D\) planer region.

Q.4. What is a Parallelogram’s Perimeter?
Ans: All the sides are added together to determine the perimeter of a parallelogram. The formula to calculate the perimeter of any parallelogram is
Perimeter = \(2 (a + b)\)

Q.5. What is the formula for finding the area of a parallelogram?
Ans: There are various methods to find the area of a parallelogram.
(i) First formula: In terms of height and base
Area of a parallelogram \( = \) base \( \times \) height
(ii) Second formula: In terms of diagonals and the angle between them
Area of a parallelogram \( = \;\frac{1}{2} \times \;p\; \times \;q\times\sin\;\left( x \right)\) where, \(‘p’\) and \(‘q’\) are diagonal and \(‘x’\) is the measure of the angle formed where the diagonals bisect.

Q.6. How to find the area of an irregular parallelogram?
Ans: Measure all the sides of the given irregular shape but before that make sure all the sides should be in the same units. Draw the area on the piece of paper using the measurements obtained. Later divide the drawings into different shapes like squares and rectangles. Deduce the area of the shapes obtained and then add areas of the individual shape.

Now that you are provided with all the necessary information on the area of a parallelogram and we hope this article on the area of the parallelogram has helped you. If you have any questions feel to post your comment below. We will get back to you at the earliest.