Area Of Parallelogram With Coordinates

Area 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 quadrilateral case, where reverse sides are equal and parallel. The surface area of a parallelogram is the infinite enclosed inside its four sides. The area of a parallelogram is equal to the product of length and summit 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 two-dimensional figure, information technology has an area and perimeter. In this commodity, allow us discuss the area of a parallelogram with its formula, derivations, and more solved problems in detail.

Too check:Mathematics Solutions

Tabular array of contents:

Definition
Formula
How to Calculate
- Using Sides
- Without Height
- Using Diagonals
Example Questions
Discussion Problem
FAQs

What is the Area of Parallelogram?

The expanse of a parallelogram is the region divisional past the parallelogram in a given 2-dimension space. To think, a parallelogram is a special type of quadrilateral having the pair of opposite sides are parallel. In a parallelogram, the opposite sides are of equal length and reverse angles are of equal measures.

Area of a Parallelogram

Area of Parallelogram Formula

To find the area of the parallelogram, multiply the base of operations of the perpendicular by its acme. 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 drawn to represent the height.

Area of Parallelogram Formula

Therefore,

Expanse = b × h Square units

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

Let the states learn the derivation of the area of a parallelogram, in the adjacent section.

How to Calculate the Expanse of Parallelogram?

The parallelogram area tin can be calculated using its base and peak. Apart from information technology, the area of a parallelogram tin also be evaluated if its ii diagonals are known along with whatever of their intersecting angles or if the length of the parallel sides is known, along with whatsoever of the angles between the sides. Hence, there are three methods to derive the expanse of a parallelogram:

When base of operations and height of the parallelogram are given
When height is not given
When diagonals are given

How to Calculate the Area of Parallelograms?

Area of Parallelogram Using Sides

Suppose a and b are the set of parallel sides of a parallelogram and h is the height, and so based on the length of sides and height of information technology, the formula for its area is given past:

Area = Base × Height

A = b × h [sq.unit]

Example: If the base of a parallelogram is equal to five cm and the acme is 3 cm, and then detect its area.

Solution: Given, the length of base=five cm and height = three cm

Equally per the formula, Expanse = five × 3 = 15 sq.cm

Expanse of Parallelogram Without Height

If the height of the parallelogram is unknown to us, then nosotros can employ the trigonometry concept hither to notice its expanse.

Expanse = ab sin (x)

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

Case: The angle betwixt any two sides of a parallelogram is 90 degrees. If the length of the two adjacent sides are 3 cm and 4 cm, respectively, then observe the area.

Solution: Let a = iii cm and b=four cm

x = 90 degrees

Area = ab sin (x)

A = 3 × 4 sin (90)

A = 12 sin ninety

A = 12 × ane = 12 sq.cm.

Notation: If the bending betwixt the sides of a parallelogram is 90 degrees, and so it is a rectangle.

Area of Parallelogram Using Diagonals

The area of any parallelogram tin can also be calculated using its diagonal lengths. Every bit we know, there are two diagonals for a parallelogram, which intersect each other. Suppose the diagonals intersect each other at an angle y, so the area of the parallelogram is given by:

Area = ½ × d₁ × d_ii sin (y)

Check the tabular array below to become summarised formulas of an area of a parallelogram.

All Formulas to Calculate Area of a Parallelogram
Using Base and Pinnacle	A = b × h
Using Trigonometry	A = ab sin (10)
Using Diagonals	A = ½ × d₁ × d_ii sin (y)

Where,

b = base of operations of the parallelogram (AB)
h = top of the parallelogram
a = side of the parallelogram (AD)
x = any angle between the sides of the parallelogram (∠DAB or ∠ADC)
d₁ = diagonal of the parallelogram (p)
d_two = diagonal of the parallelogram (q)
y = any angle between at the intersection indicate of the diagonals (∠DOA or ∠DOC)

Note: In the higher up figure,

DC = AB = b
AD = BC = a
∠DAB = ∠DCB
∠ADC = ∠ABC
O is the intersecting point of the diagonals
∠DOA = ∠COB
∠Dr. = ∠AOB

Area of Parallelogram in Vector Form

If the sides of a parallelogram are given in vector form, and so the area of the parallelogram tin can exist 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 the parallelogram.

Area of a parallelogram in vector grade = Mod of cross-production of vector a and vector b

A = | a × b|

At present, we accept to find the area of a parallelogram with respect to diagonals, say d₁ and d₂, in vector form.

And so, we can write;

a + b = d₁

b + (-a) = d_ii

b – a = d₂

Thus,

d₁ × d₂ = (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,

d₁ × d₂ = a × b + a × b = 2 (a × b)

a × b = one/2 (d₁ × d_two)

Hence,

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

A = 1/2 (d_one × d_two)

where d1 and d2 are vectors of diagonals.

Instance: Discover the area of a parallelogram whose side by side sides are given in vectors.

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

Area of parallelogram = |A × B|

\(\begin{assortment}{l}= \begin{vmatrix} i & j & grand\\ 3 & 2 & 0\\ -3 & 1 & 0 \cease{vmatrix}\stop{assortment} \)

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

= 9k

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

Area of a Circumvolve
Surface area of a Triangle
Expanse of Foursquare
Areas Of Parallelograms And Triangles Class 9

Solved Examples on Area of Parallelogram

Question 1: Find the area of the parallelogram with a base of operations of iv cm and acme of five cm.

Solution:

Given:

Base of operations, b = four cm

h = five cm

We know that,

Expanse of Parallelogram = b × h Square units

= four × 5 = 20 sq.cm

Therefore, the area of a parallelogram = xx cm²

Question 2: Find the area of a parallelogram whose breadth is viii cm and height is 11 cm.

Solution:

Given,

b = 8 cm

h = xi cm

Expanse of a parallelogram

= b × h

= 8 × xi cmⁱⁱ

= 88 cmⁱⁱ

Question 3: The base of the parallelogram is thrice its height. If the area is 192 cm^two, discover the base and height.

Solution:

Allow the peak of the parallelogram = h cm

then, the base of operations of the parallelogram = 3h cm

Area of the parallelogram = 192 cm²

Surface area of parallelogram = base × height

Therefore, 192 = 3h × h

⇒ 3 × h² = 192

⇒ hⁱⁱ = 64

⇒ h = viii cm

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

3 × h

= 3 × viii

= 24 cm

Discussion Problem on Area of Parallelogram

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

Solution:

Given, area = 500 sq.cm.

Tiptop = Twice of base of operations

h = 2b

By the formula, we know,

Surface area = b x h

500 = b x 2b

2b² = 500

bⁱⁱ = 250

b = 15.eight cm

Hence, summit = 2 x b = 31.6 cm

Practise Questions on Area of a Parallelogram

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

Frequently Asked Questions

What is a Parallelogram?

A parallelogram is a geometrical figure that has 4 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 surface area of any parallelogram can be calculated using the following formula:

Expanse = base × peak

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

What is the Perimeter of a Parallelogram?

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

Perimeter = 2 (a + b)

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

The area of a perpendicular with peak 5 cm and base iv cm will be;

A = b × h

Or, A = 4 × v = 20 cmⁱⁱ