How Do You Find the Area of a Parallelogram Without Knowing the Height or Base

Surface area of a parallelogram is a region covered by a parallelogram in a 2-dimensional airplane. In Geometry, a parallelogram is a two-dimensional effigy with 4 sides. It is a special instance of the quadrilateral, where contrary sides are equal and parallel. The area of a parallelogram is the infinite enclosed within its four sides. Area is equal to the product of length and height of the parallelogram.

The sum of the interior angles in a quadrilateral is 360 degrees. A parallelogram has ii pairs of parallel sides with equal measures. Since it is a ii-dimensional figure, information technology has an area and perimeter. In this commodity, let us discuss the area of a parallelogram with its formula, derivations, and more solved problems in item.

Also check:Mathematics Solutions

Table of contents:

• Definition
• Formula
• How to Calculate
  • Using Sides
  • Without Meridian
  • Using Diagonals
• Case Questions
• Word Problem
• FAQs

What is the Surface 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 4 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 expanse of the parallelogram, multiply the base of the perpendicular by its height. 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 summit.

Therefore,

Area = b × h Square units

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

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

How to Summate the Expanse of Parallelogram?

The parallelogram area can be calculated, using its base and tiptop. Apart from information technology, the area 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 any of the angles between the sides. Hence, there 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

Surface area of Parallelogram Using Sides

Suppose a and b are the set up 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 × Peak

A = b × h     [sq.unit of measurement]

Instance: If the base of operations of a parallelogram is equal to five cm and the elevation is 3 cm, and then find its area.

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

Equally per the formula, Expanse = 5 × 3 = xv sq.cm

Area of Parallelogram Without Height

If the summit of the parallelogram is unknown to us, then nosotros can utilize the trigonometry concept here to find 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.

Example: The angle between whatever two sides of a parallelogram is 90 degrees. If the length of the ii parallel sides is iii cm and four cm respectively, and so find the area.

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

x = 90 degrees

Area = ab sin (x)

A = 3 × 4 sin (ninety)

A = 12 sin 90

A = 12 × one = 12 sq.cm.

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

Surface 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 ii diagonals for a parallelogram, which intersects each other. Suppose, the diagonals intersect each other at an angle y, then the expanse of the parallelogram is given past:

Area = ½ × d_one × d_two sin (y)

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

All Formulas to Calculate Surface area of a Parallelogram
Using Base of operations and Tiptop	A = b × h
Using Trigonometry	A = ab sin (x)
Using Diagonals	A = ½ × d1 × d2 sin (y)

Where,

• b = base of the parallelogram (AB)
• h = height of the parallelogram
• a = side of the parallelogram (Advertizing)
• x = any angle between the sides of the parallelogram (∠DAB or ∠ADC)
• d₁ = diagonal of the parallelogram (p)
• d_ii = diagonal of the parallelogram (q)
• y = whatever angle between at the intersection signal of the diagonals (∠DOA or ∠DOC)

Annotation: In the above effigy,

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

Surface area of Parallelogram in Vector Form

If the sides of a parallelogram are given in vector form then the expanse of the parallelogram can exist calculated using its diagonals. Suppose, vector 'a' and vector 'b' are the ii sides of a parallelogram, such that the resulting vector is the diagonal of parallelogram.

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

A = | a × b|

Now, nosotros have to find the area of a parallelogram with respect to diagonals, say d₁ and d₂,  in vector form.

So, nosotros can write;

a + b = d₁

b + (-a) = d₂

or

b – a = d₂

Thus,

d₁ × d_ii = (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 = 1/2 (d₁ × d₂)

Hence,

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

A = 1/2 (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

Surface area of parallelogram = |A × B|

= \(\begin{vmatrix} i & j & k\\ 3 & ii & 0\\ -3 & one & 0 \terminate{vmatrix}\)

= i (0-0) – (0-0) + k(3+6)  [Using determinant of 3 10 3 matrix formula]

= 9k

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

Related Manufactures

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

Solved Examples on Area of Parallelogram

Question ane: Discover the area of the parallelogram with the base of 4 cm and summit of 5 cm.

Solution:

Given:

Base, b = 4 cm

h = 5 cm

We know that,

Area of Parallelogram = b×h Foursquare units

= 4 × 5 = 20 sq.cm

Therefore, the area of a parallelogram = 20 cmⁱⁱ

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

Solution:

Given,

b = viii cm

h = 11 cm

Area of a parallelogram

= b × h

= viii × 11 cm²

= 88 cm²

Question 3: The base of the parallelogram is thrice its peak. If the expanse is 192 cm^two, detect the base and peak.

Solution:

Allow 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 × tiptop

Therefore, 192 = 3h × h

⇒ 3 × h² = 192

⇒ h^two = 64

⇒ h = 8 cm

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

3 × h

= 3 × viii

= 24 cm

Word Problem on Expanse of Parallelogram

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

Solution:

Given, area = 500 sq.cm.

Acme = Twice of base of operations

h = 2b

By the formula, we know,

Surface area = b x h

500 = b x 2b

2b² = 500

b² = 250

b = 15.8 cm

Hence, elevation = 2 ten b = 31.half dozen cm

Practise Questions on Surface area of a Parallelogram

1. Detect the area of a parallelogram whose base is 8 cm and height is 4 cm.
2. Find the surface area of a parallelogram with a base equal to seven inches and height is nine inches.
3. The base of the parallelogram is thrice its summit. If the area is 147 sq.units, and so what is the value of its base of operations and summit?
4. A parallelogram has sides equal to 10m and 8m. If the distance between the shortest sides is 5m, then find the distance between the longest sides of the parallelogram. (Hint: Outset find the area of parallelogram using altitude between shortest sides)

Frequently 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 opposite sides are equal in length, and opposite angles are equal in measure.

What is the Area of a Parallelogram?

The area of any parallelogram can exist calculated using the post-obit formula:

Area = base × tiptop

Information technology should exist noted that the base and meridian of a parallelogram must exist perpendicular.

What is the Perimeter of a Parallelogram?

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

Perimeter = ii (a + b)

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

The area of a perpendicular with height 5 cm and base 4 cm will exist;

A = b × h

Or, A = iv × 5 = 20 cm²

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