How To Find The Area Of A Parallelogram How To Find The Area Of A Triangle

Expanse of a parallelogram is a region covered by a parallelogram in a 2-dimensional plane. In Geometry, a parallelogram is a ii-dimensional figure with four sides. It is a special case of the quadrilateral, where opposite sides are equal and parallel. The surface area of a parallelogram is the space enclosed within its four sides. Surface area is equal to the production of length and pinnacle of the parallelogram.

The sum of the interior angles in a quadrilateral is 360 degrees. A parallelogram has two pairs of parallel sides with equal measures. Since it is a ii-dimensional effigy, it has an expanse and perimeter. In this commodity, permit us discuss the area of a parallelogram with its formula, derivations, and more solved issues in detail.

Also cheque:Mathematics Solutions

Table of contents:

• Definition
• Formula
• How to Summate
  • Using Sides
  • Without Height
  • Using Diagonals
• Example Questions
• Word Problem
• FAQs

What is the Area of Parallelogram?

The area of a parallelogram is the region bounded by the parallelogram in a given two-dimension space. To recall, a parallelogram is a special type of quadrilateral which has four sides and the pair of opposite sides are parallel. In a parallelogram, the opposite sides are of equal length and reverse angles are of equal measures. Since the rectangle and the parallelogram take similar properties, the area of the rectangle is equal to the area of a parallelogram.

Area of Parallelogram Formula

To find the area of the parallelogram, multiply the base of the perpendicular past its meridian. It should be noted that the base and the height of the parallelogram are perpendicular to each other, whereas the lateral side of the parallelogram is not perpendicular to the base. Thus, a dotted line is fatigued to represent the height.

Therefore,

Surface area = b × h Square units

Where "b" is the base and "h" is the height of the parallelogram.

Let us learn the derivation of area of a parallelogram, in the next section.

How to Summate the Surface area of Parallelogram?

The parallelogram expanse tin be calculated, using its base and height. Apart from it, the expanse of a parallelogram can also be evaluated, if its two diagonals are known along with whatsoever of their intersecting angles, or if the length of the parallel sides is known, along with whatsoever of the angles between the sides. Hence, at that place are three method to derive the area of parallelogram:

• When base and height of parallelogram are given
• When tiptop is not given
• When diagonals are given

Area of Parallelogram Using Sides

Suppose a and b are the set of parallel sides of a parallelogram and h is the height, then based on the length of sides and height of it, the formula for its area is given by:

Area = Base of operations × Height

A = b × h     [sq.unit]

Case: If the base of operations of a parallelogram is equal to 5 cm and the height is iii cm, then observe its area.

Solution: Given, length of base=5 cm and height = 3 cm

As per the formula, Area = 5 × iii = fifteen sq.cm

Area of Parallelogram Without Height

If the acme of the parallelogram is unknown to us, then we can use the trigonometry concept here to notice its area.

Area = ab sin (x)

Where a and b are the length of parallel sides and x is the angle betwixt the sides of the parallelogram.

Case: The angle betwixt any 2 sides of a parallelogram is 90 degrees. If the length of the 2 parallel sides is 3 cm and 4 cm respectively, and so observe the area.

Solution: Let a = 3 cm and b=4 cm

10 = 90 degrees

Expanse = ab sin (x)

A = iii × 4 sin (xc)

A = 12 sin 90

A = 12 × 1 = 12 sq.cm.

Annotation: If the bending between the sides of a parallelogram is xc degrees, and so it is a rectangle.

Area of Parallelogram Using Diagonals

The area of any parallelogram can likewise be calculated using its diagonal lengths. As we know, there are two diagonals for a parallelogram, which intersects each other. Suppose, the diagonals intersect each other at an angle y, then the surface area of the parallelogram is given by:

Area = ½ × d₁ × d₂ sin (y)

Check the table below to get summarised formulas of an area of a parallelogram.

All Formulas to Calculate Surface area of a Parallelogram
Using Base of operations and Height	A = b × h
Using Trigonometry	A = ab sin (ten)
Using Diagonals	A = ½ × d1 × dtwo sin (y)

Where,

• b = base of the parallelogram (AB)
• h = height of the parallelogram
• a = side of the parallelogram (AD)
• x = any bending between the sides of the parallelogram (∠DAB or ∠ADC)
• d_i = diagonal of the parallelogram (p)
• d₂ = diagonal of the parallelogram (q)
• y = any angle between at the intersection point of the diagonals (∠DOA or ∠DOC)

Note: In the above figure,

• DC = AB = b
• Advert = BC = a
• ∠DAB = ∠DCB
• ∠ADC = ∠ABC
• O is the intersecting signal of the diagonals
• ∠DOA = ∠COB
• ∠DOC = ∠AOB

Area of Parallelogram in Vector Form

If the sides of a parallelogram are given in vector course and then the surface area of the parallelogram can be calculated using its diagonals. Suppose, vector 'a' and vector 'b' are the two sides of a parallelogram, such that the resulting vector is the diagonal of parallelogram.

Area of parallelogram in vector class = Mod of cross-product of vector a and vector b

A = | a × b|

Now, nosotros have to find the expanse of a parallelogram with respect to diagonals, say d_i and d₂,  in vector course.

Then, we can write;

a + b = d_one

b + (-a) = d₂

or

b – a = d₂

Thus,

d₁ × d_two = (a + b) × (b – a)

= a × (b – a) + b × (b – a)

= a × b – a × a + b × b – b × a

= a × b – 0 + 0 – b × a

= a × b – b × a

Since,

a × b = – b × a

Therefore,

d_i × d_ii = a × b + a × b = 2 (a × b)

a × b = one/2 (d₁ × d_ii)

Hence,

Area of parallelogram when diagonals are given in the vector form, becomes:

A = 1/two (d₁ × d₂)

where d1 and d2 are vectors of diagonals.

Example: Find the area of parallelogram whose adjacent sides are given in vectors.

A = 3i + 2j and B = -3i + 1j

Expanse of parallelogram = |A × B|

=

\(\brainstorm{array}{l}\brainstorm{vmatrix} i & j & m\\ iii & ii & 0\\ -iii & 1 & 0 \cease{vmatrix}\stop{array} \)

= i (0-0) – (0-0) + k(3+six)  [Using determinant of three x 3 matrix formula]

= 9k

Thus, expanse of the parallelogram formed by two vectors A and B is equal to 9k sq.unit.

Related Articles

• Area of a Circle
• Surface area of a Triangle
• Area of Square
• Areas Of Parallelograms And Triangles Grade ix

Solved Examples on Area of Parallelogram

Question ane: Find the area of the parallelogram with the base of 4 cm and meridian of 5 cm.

Solution:

Given:

Base of operations, b = four cm

h = five cm

We know that,

Surface area of Parallelogram = b×h Square units

= 4 × 5 = twenty sq.cm

Therefore, the area of a parallelogram = 20 cm^two

Question 2: Find the expanse of a parallelogram whose breadth is 8 cm and height is eleven cm.

Solution:

Given,

b = 8 cm

h = 11 cm

Area of a parallelogram

= b × h

= eight × 11 cm²

= 88 cm²

Question 3: The base of the parallelogram is thrice its height. If the area is 192 cm², find the base and height.

Solution:

Permit the height of the parallelogram = h cm

then, the base of the parallelogram = 3h cm

Area of the parallelogram = 192 cm²

Area of parallelogram = base × height

Therefore, 192 = 3h × h

⇒ three × h^two = 192

⇒ h² = 64

⇒ h = 8 cm

Hence, the height of the parallelogram is 8 cm, and breadth is

3 × h

= 3 × 8

= 24 cm

Word Problem on Area of Parallelogram

Question: The expanse of a parallelogram is 500 sq.cm. Its height is twice its base. Find the peak and base.

Solution:

Given, surface area = 500 sq.cm.

Summit = Twice of base

h = 2b

By the formula, we know,

Area = b x h

500 = b x 2b

2bⁱⁱ = 500

b^two = 250

b = fifteen.eight cm

Hence, top = 2 x b = 31.half dozen cm

Practise Questions on Expanse of a Parallelogram

1. Find the area of a parallelogram whose base of operations is 8 cm and acme is four cm.
2. Find the area of a parallelogram with a base of operations equal to 7 inches and pinnacle is 9 inches.
3. The base of operations of the parallelogram is thrice its height. If the surface area is 147 sq.units, then what is the value of its base and height?
4. A parallelogram has sides equal to 10m and 8m. If the distance betwixt the shortest sides is 5m, then find the distance between the longest sides of the parallelogram. (Hint: Commencement find the expanse of parallelogram using altitude betwixt shortest sides)

Often Asked Questions

What is a Parallelogram?

A parallelogram is a geometrical figure that has four sides formed by two pairs of parallel lines. In a parallelogram, the reverse sides are equal in length, and reverse angles are equal in measure.

What is the Expanse of a Parallelogram?

The area of any parallelogram can exist calculated using the following formula:

Surface area = base × height

It should be noted that the base and acme of a parallelogram must exist perpendicular.

What is the Perimeter of a Parallelogram?

To observe the perimeter of a parallelogram, add all the sides together. The following formula gives the perimeter of any parallelogram:

Perimeter = 2 (a + b)

What is the Area of a Parallelogram whose height is 5 cm and base is 4 cm?

The expanse of a perpendicular with summit v cm and base of operations four cm will be;

A = b × h

Or, A = iv × 5 = twenty cm²

Source: https://byjus.com/maths/area-of-parallelogram/

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