Area of Parallelogram - Formula, Definition, Examples (2024)

The area of a parallelogram is defined as the region or space covered by a parallelogram in a two-dimensional plane. A parallelogram is a special kind of quadrilateral. If a quadrilateral has two pairs of parallel opposite sides, then it is called a parallelogram. Rectangle, square, and rhombus are all examples of a parallelogram. Geometry is all about shapes, 2D or 3D. All of these shapes have a different set of properties with different formulas for area. The prime focus here will be entirely on the following:

  • Definition of the area of a parallelogram
  • Formula of the area of a parallelogram
  • Calculation of a parallelogram's area in vector form
1.What Is the Area of Parallelogram?
2.Area of Parallelogram Formula
3.How To Calculate the Area of Parallelogram?
4.Area of Parallelogram in Vector Form
5.FAQs on Area of Parallelogram

What Is the Area of Parallelogram?

The area of a parallelogram refers to the total number of unit squares that can fit into it and it is measured in square units (like cm2, m2, in2, etc). It is the region enclosed or encompassed by a parallelogram in two-dimensional space. Let us recall the definition of a parallelogram. A parallelogram is a four-sided, 2-dimensional figure with:

  • two equal, opposite sides,
  • two intersecting and non-equal diagonals, and
  • opposite angles that are equal

We come across many geometric shapes other than rectangles and squares in our daily lives. Since few properties of a rectangle and the parallelogram are somewhat similar, the area of the rectangle is similar to the area of a parallelogram.

Area of a Parallelogram Formula

The area of a parallelogram can be calculated by multiplying its base with the altitude. The base and altitude of a parallelogram are perpendicular to each other as shown in the following figure. The formula to calculate the area of a parallelogram can thus be given as,

Area of parallelogram = b × h square units
where,

  • b is the length of the base
  • h is the height or altitude

Area of Parallelogram - Formula, Definition, Examples (1)

Let us analyze the above formula using an example. Assume that PQRS is a parallelogram. Using grid paper, let us find its area by counting the squares.

Area of Parallelogram - Formula, Definition, Examples (2)

From the above figure:
Total number of complete squares = 16
Total number of half squares = 8
Area = 16 + (1/2) × 8 = 16 + 4 = 20 unit2

Also, we observe in the figure that ST ⊥ PQ. By counting the squares, we get:
Side, PQ = 5 units
Corresponding height, ST=4 units
Side × height = 5 × 4 = 20 unit2

Thus, the area of the given parallelogram is base times the altitude.

Let's do an activity to understand the area of a parallelogram.

  • Step I: Draw a parallelogram (PQRS) with altitude (SE) on a cardboard and cut it.
  • Step II: Cut the triangular portion (PSE).
  • Step III: Paste the remaining portion (EQRS) on a white chart.
  • Step IV: Paste the triangular portion (PSE) on the white chart joining sides RQ and SP.

Area of Parallelogram - Formula, Definition, Examples (3)

After doing this activity, we observed that the area of a rectangle is equal to the area of a parallelogram. Also, the base and height of the parallelogram are equal to the length and breadth of the rectangle respectively.

Area of Parallelogram = Base × Height

How To Calculate Area of a Parallelogram?

The parallelogram area can be calculated with the help of its base and height. Also, the area of a parallelogram can also be evaluated if its two diagonals along with any of their intersecting angles are known, or if the length of the parallel sides along with any of the angles between the sides is known.

Parallelogram Area Using Height

Suppose 'a' and 'b' are the set of parallel sides of a parallelogram and 'h' is the height (which is the perpendicular distance between 'a' and 'b'), then the area of a parallelogram is given by:

Area = Base × Height

A = b × h [square units]

Example: If the base of a parallelogram is equal to 5 cm and the height is 4 cm, then find its area.

Solution: Given, length of base = 5 cm and height = 4 cm

As per the formula, Area = 5 × 4 = 20 cm2

Parallelogram Area Using Lengths of Sides

The area of a parallelogram can also be calculated without the height if the length of adjacent sides and angle between them are known to us. We can simply use the area of the triangle formula from the trigonometry concept for this case.

Area = ab sin (θ)

where,

  • a and b = length of parallel sides, and,
  • θ = angle between the sides of the parallelogram.

Example: The angle between any two sides of a parallelogram is 90 degrees. If the length of the two parallel sides is 4 units and 6 units respectively, then find the area.

Solution:

Let a = 4 units and b = 6 units
θ = 90 degrees

Using area of parallelogram formula,
Area = ab sin (θ)
⇒ A = 4 × 6 sin (90º)
⇒ A = 24 sin 90º
⇒ A = 24 × 1 = 24 sq.units.

Note: If the angle between the sides of a parallelogram is 90 degrees, then the parallelogram becomes a rectangle.

Parallelogram Area Using Diagonals

The area of any given parallelogram can also be calculated using the length of its diagonals. There are two diagonals for a parallelogram, intersecting each other at certain angles. Suppose, this angle is given by x, then the area of the parallelogram is given by:

Area = ½ × d\(_1\) × d\(_2\) sin (x)

where,

  • d\(_1\) and d\(_2\) = Length of diagonals of the parallelogram, and
  • x = Angle between the diagonals.

Area of Parallelogram in Vector Form

The area of the parallelogram can be calculated using different formulas even when either the sides or the diagonals are given in the vector form. Consider a parallelogram ABCD as shown in the figure below,

Area of Parallelogram - Formula, Definition, Examples (4)

Area of parallelogram in vector form using the adjacent sides is,

\(|\overrightarrow{\mathrm{a}} × \overrightarrow{\mathrm{b}}|\)
where, \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are vectors representing two adjacent sides.

Here,
\(\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{d_1} \) → i) and,
\(\overrightarrow{b} + (-\overrightarrow{a}) = \overrightarrow{d_2} \)
or, \(\overrightarrow{b} - \overrightarrow{a} = \overrightarrow{d_2}\) → ii)

⇒ \( \overrightarrow{d_1} \times \overrightarrow{d_2} = (\overrightarrow{a} + \overrightarrow{b}) (\overrightarrow{b} - \overrightarrow{a})\)
= \(\overrightarrow{a}\) × (\(\overrightarrow{b}\) - \(\overrightarrow{a}\)) + \(\overrightarrow{b}\) × (\(\overrightarrow{b}\) - \(\overrightarrow{a}\))
= \(\overrightarrow{a}\) × \(\overrightarrow{b}\) - \(\overrightarrow{a}\) × \(\overrightarrow{a}\) + \(\overrightarrow{b}\) × \(\overrightarrow{b}\) - \(\overrightarrow{b}\) × \(\overrightarrow{a}\)

Since \(\overrightarrow{a}\) × \(\overrightarrow{a}\) = 0, and \(\overrightarrow{b}\) × \(\overrightarrow{b}\) = 0
⇒ \(\overrightarrow{a}\) × \(\overrightarrow{b}\) - 0 + 0 - \(\overrightarrow{b}\) × \(\overrightarrow{a}\)

Since \(\overrightarrow{a}\) × \(\overrightarrow{b}\) = - \(\overrightarrow{b}\) × \(\overrightarrow{a}\),
\( \overrightarrow{d_1}\) × \(\overrightarrow{d_2}\) = \(\overrightarrow{a}\) × \(\overrightarrow{b}\) - (-(\(\overrightarrow{a}\) × \(\overrightarrow{b}\)))
= 2(\(\overrightarrow{a}\) × \(\overrightarrow{b}\))

Therefore, area of parallelogram when diagonals are given in vector form = 1/2 |(\(\overrightarrow{d_1}\) × \(\overrightarrow{d_2}\))|
where, \(\overrightarrow{d_1}\) and \(\overrightarrow{d_2}\) are diagonals.

Thinking Out Of the Box!

  • Can a kite be called a parallelogram?
  • What elements of a trapezoid should be changed to make it a parallelogram?
  • Can there be a concave parallelogram?
  • Can you find the area of the parallelogram without knowing its height?

FAQs on Area of Parallelogram

What Is the Area of a Parallelogram in Math?

The area of a parallelogram is defined as the region enclosed or encompassed by a parallelogram in two-dimensional space. It is represented in square units like cm2, m2, in2, etc.

How To Find Area of a Parallelogram Without Height?

The area of a parallelogram can be calculated without the height when the length of adjacent sides and the angle between them is known. The formula to find the area for this case is given as, area = ab sin (θ), where 'a' and 'b' are lengths of adjacent sides, and, θ is the angle between them.

Also, the area can be calculated when the diagonals and their intersecting angle are given, using the formula, Area = ½ × d1 × d2 sin (y), where 'd1' and 'd2' are lengths of diagonals of the parallelogram, and 'y' is the angle between them.

What Is the Formula of Finding Area of Parallelogram?

The area of a parallelogram can be calculated by finding the product of its base with the altitude. The base and altitude of a parallelogram are always perpendicular to each other. The formula to calculate the area of a parallelogram is given as Area of parallelogram = base × height square units.

How To Find Area of Parallelogram With Vectors?

Area of a parallelogram can be calculated when the adjacent sides or diagonals are given in the vector form. The formula to find area using vector adjacent sides is given as, | \(\overrightarrow{a}\) × \(\overrightarrow{b}\)|, where \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are adjacent side vectors. Also, the area of parallelogram formula using diagonals in vector form is, area = 1/2 |(\(\overrightarrow{d_1}\) × \(\overrightarrow{d_2}\))|, where \(\overrightarrow{d_1}\) and \(\overrightarrow{d_2}\) are diagonal vectors.

How to Calculate Area of Parallelogram Using Calculator?

To determine the area of a parallelogram the easiest and fastest method is to use the area of a parallelogram calculator. It is a free online tool that helps you to calculate the area of a parallelogram with the help of the given dimensions. Try now Cuemath's area of parallelogram calculator, enter the value of height and base of the parallelogram and get the parallelogram's area within a few seconds.

What Is the Area of a Parallelogram When Diagonals are Given?

The area of a parallelogram can be calculated when the diagonals and their intersecting angle are known. The formula is given as, area = ½ × d1 × d2 sin (x), where 'd1' and 'd2' are lengths of diagonals of the parallelogram, and 'x' is the angle between them.

How To Calculate Area of Parallelogram Whose Adjacent Sides are Given?

To find the area of a parallelogram when the lengths of adjacent sides are given, we need the angle between them. The formula to find the area for this case is given as, area = ab sin (θ), where 'a' and 'b' are lengths of adjacent sides, and, θ is the angle between the sides of the parallelogram.

Area of Parallelogram - Formula, Definition, Examples (2024)

FAQs

What is the area formula for parallelograms? ›

The area of a parallelogram can be calculated by finding the product of its base with the altitude. The base and altitude of a parallelogram are always perpendicular to each other. The formula to calculate the area of a parallelogram is given as Area of parallelogram = base × height square units.

What is the definition and example of a parallelogram? ›

What is a Parallelogram in Geometry? In geometry, a quadrilateral is called a parallelogram. A parallelogram has its opposite sides parallel and equal in length. Few examples of a parallelogram are rhombus, rectangle, and square.

What is the best definition of a parallelogram? ›

A parallelogram is a special type of quadrilateral that has both pairs of opposite sides parallel and equal.

How do you find area? ›

How to Find the Area of a Square or Rectangle - YouTube

What are the 4 types of parallelograms? ›

There are 4 types of parallelograms, including 3 special types. The four types are parallelograms, squares, rectangles, and rhombuses.

What is a parallelogram in maths? ›

A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle.

What is a parallelogram Class 8? ›

Basically, a parallelogram is a quadrilateral whose two sides are parallel. The opposite sides and the opposite angles of a parallelogram are also equal.

How do you solve a parallelogram problem? ›

Let the lengths of two sides of the parallelogram be 4x cm and 3x cm respectively. Then, its perimeter = 2(4x + 3x) cm = 8x + 6x = 14x cm. Therefore, 14x = 56 ⇔ x = ⁵⁶/₁₄ = 4. Therefore, one side = (4 × 4) cm = 16 cm and other side = (3 × 4) cm = 12 cm.

What is the best definition of a parallelogram Brainly? ›

Answer: quadrilateral with two pairs of parallel sides.

What is a parallelogram for kids? ›

A parallelogram is a shape with four sides, and the sides opposite each other are parallel, meaning they don't intersect. Examples of parallelograms include squares, rhombuses, and rectangles. Circles, triangles, and trapezoids are not parallelograms.

What is the best definition of a parallelogram Quizizz? ›

any four sided figure with opposite sides congruent.

What is the formula of area of? ›

Area and Perimeter Formula Chart
FiguresArea FormulaVariables
Area of SquareArea = a2a = sides of the square
Area of a TriangleArea = 1/2 b×hb = base h = height
Area of a CircleArea = πr2r = radius of the circle
Area of a TrapezoidArea = 1/2 (a + b)ha =base 1 b = base 2 h = vertical height
2 more rows
30 Sept 2020

What is area of a shape? ›

The area of a shape is the “space enclosed within the perimeter or the boundary” of the given shape. We calculate the area for different shapes using math formulas.

What is a formula of area of rectangle? ›

Area of a Rectangle. A = l × b. The area of any rectangle is calculated, once its length and width are known. By multiplying length and breadth, the rectangle's area will obtain in a square-unit dimension.

Is parallelogram a square? ›

A square is a parallelogram. This is always true. Squares are quadrilaterals with 4 congruent sides and 4 right angles, and they also have two sets of parallel sides. Parallelograms are quadrilaterals with two sets of parallel sides.

Do parallelograms have 4 equal sides? ›

What is a parallelogram? A parallelogram is a quadrilateral with 2 pairs of parallel sides. In these figures, sides of the same color are parallel to each other. A shape with four sides of equal length.

Can a parallelogram have 6 sides? ›

A parallelogram has six sides. Opposite sides of a parallelogram are parallel. A parallelogram can have one pair or two pairs of parallel sides. All sides of a parallelogram have the same length.

What is the formula for area of quadrilateral? ›

Area of quadrilateral = (½) × diagonal length × sum of the length of the perpendiculars drawn from the remaining two vertices.

How do you find the surface area of a parallelogram prism? ›

The formula for the surface area of a prism is obtained by taking the sum of (twice the base area) and (the lateral surface area of the prism). The surface area of a prism is given as S = (2 × Base Area) + (Base perimeter × height) where "S" is the surface area of the prism.

How do you find the area of a parallelogram with 4 points? ›

Area Of A Parallelogram Using Determinants - Linear Algebra ...

What is the area of parallelogram in vector? ›

Since, the sine component of the vector resembles the height of the parallelogram made by the vectors. Hence, the magnitude of the cross product is the area of the parallelogram. Hence, the area of the parallelogram will be \[27sq. \,unit\].

What is a formula of perimeter of parallelogram? ›

According to the property of the parallelogram, the opposite sides are parallel to each other, and the parallelogram perimeter is defined as two times of the base and height. Thus, the formula for the perimeter of a parallelogram is: P = 2 (b +h/cos θ)

What is quadrilateral and its formula? ›

Five different formulas are used to calculate the area of the quadrilateral.
...
Area Formulas of Quadrilaterals.
Quadrilateral Area Formulas
Area of a Kite(1 ⁄ 2) × Product of Diagonals
Area of a ParallelogramBase × Height
Area of a RectangleLength × Breadth
Area of a Trapezoidb a s e 1 + b a s e 2 2 × h e i g h t
1 more row

What is the prism formula? ›

The Prism Formula is as follows, The surface area of a prism = (2×BaseArea) +Lateral Surface Area. The volume of a prism =Base Area× Height.

What is the area of prism? ›

The formula for the surface area of a prism is SA=2B+ph, where B, again, stands for the area of the base, p represents the perimeter of the base, and h stands for the height of the prism.

What is volume and surface area? ›

The surface area of any given object is the area or region occupied by the surface of the object. Whereas volume is the amount of space available in an object. In geometry, there are different shapes and sizes such as sphere, cube, cuboid, cone, cylinder, etc. Each shape has its surface area as well as volume.

Why does the formula for area of a parallelogram work? ›

Because base × height gives the area of the rectangle, we can use the same measurements on the parallelogram to compute its area: base × height.

How do you find the area of a parallelogram with points? ›

Area of parallelogram on coordinate plane - YouTube

How do you find the area of a parallelogram with 3 numbers? ›

How to Find the Area of a Parallelogram (with 3 numbers)

How do you find the area of vectors? ›

Area of Triangle Formed by Two Vectors using Cross Product

What is law of parallelogram of vector addition? ›

Parallelogram Law of Vectors

If two vectors are acting simultaneously at a point, then it can be represented both in magnitude and direction by the adjacent sides drawn from a point.

Is area a vector or scalar? ›

Area is a vector quantity.

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