Area of Parallelogram
Area of Parallelogram: In geometry, the parallelogram is the enclosed area, a simple quadrilateral whose opposite sides are equal and whose opposite angles are equal. A parallelogram is defined as a quadrilateral that is bounded by its four sides. A Parallelogram can be of three types: square, rectangle, and rhombus. Often you would have seen parallelogram-shaped objects in your surroundings such as roofs of huts, tables, windows, erasers, etc. So this article we will discuss an area of parallelograms, their formula, and some examples.
What is Parallelogram?
In geometry, the parallelogram is the enclosed area, a simple quadrilateral whose opposite sides are equal and whose opposite angles are equal. There are some special properties of parallelograms, these are-
- Their opposite sides are parallel.
- Their Opposite sides are equal.
- Their Opposite angles are equal.
- The diagonals bisect each other.
- Their interior angle is supplementary to each other.
What is the Area of Parallelogram?
The area of a parallelogram is the region that is bounded by the parallelogram in a given two-dimension space. A particular type of quadrilateral has a pair of opposite parallel sides. Mathematically, the areas of a parallelogram are equal to the product of the base and height of the parallelogram.
Area of Parallelogram Formula
In order to calculate the area of a parallelogram, multiply the base of the parallelogram by the height of the parallelogram. The height and base are perpendicular to each other. So the formula for the area of a parallelogram is-
Area of Parallelogram = b × h square units
Where,
b = length of the base
h = height or altitude
Calculation of Area of Parallelogram
The way to calculate the area of a parallelogram is calculated with the help of base and height. Also, the area of a parallelogram can be calculated 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. so, there are three ways to calculate the area of a parallelogram.
| Calculation of Area of Parallelogram | |
| Parameters | Formula |
| Using height and base | Area = Base × Height |
| Using diagonals | Area = ab sin (θ) |
| Using the length of the sides | Area = ½ × d1 × d2 sin (x) |
Where,
- b = base of parallelogram
- h = height of a parallelogram
- a = the side of the parallelogram
- θ = angle between the sides of the parallelogram
- d1 = diagonal of the parallelogram
- d2 = diagonal of the parallelogram
- x = any angle that is between the intersection point of the diagonals.
Area of Parallelogram Examples
Question1. The base of the parallelogram is three times its height. If the area is 243 cm², find the base and height.
Solution: According to the question, the height of the parallelogram = x cm
and the base of the parallelogram is = 3x cm
Area of the parallelogram = 243cm²
As we know, the Area of the parallelogram = base × height
243 = 3x × x
⇒ 3x² = 243
⇒ x² = 81
⇒ x = 9
Therefore, 3x = 3 × 9 = 27
Therefore, the Base of the parallelogram is 27cm and the height is 9cm.
Question 2. A parallelogram has sides of 15 cm and 10 cm. If the distance between its shorter sides is 9 cm, find the distance between its longer side.
Solution: Adjacent sides of parallelogram = 15 cm and 10 cm
Distance between shorter sides = 9 cm
Area of parallelogram = b × h = 10 × 9 cm² = 80cm²
As we know, area of parallelogram = b × h
⇒ 90 = 15 × h
⇒ h = 90/15
⇒ h = 6 cm
Therefore, the distance between its longer side is 6 cm.
Question 3. Calculate the area of the roof that is spread in the shape of a parallelogram, given that, the base measures 10 in, and the altitude measures 5 in.
Solution: As we know, the area of the parallelogram formula, is = b x h
Area of the roof = b × h = (10) × (5) = 50in²
So the area of the roof is 50 in²
Question4: The area of a parallelogram is 300 sq. cm. Its height is twice its base. Find the height and base.
Solution: According to the question, the area of a parallelogram is 200 sq. cm.
And, h = 2b
So, Area = b x h
300 = b x 2b
300 = 3b
b = 300/3
b= 100 sq. cm.
The base of the parallelogram is 100 sq. cm.
Therefore, the height is 200 sq. cm.
Question 5. The angle between any two sides of a parallelogram is perpendicular to each other. If the length of the two adjacent sides is 9cm and 12cm, respectively, then find the area.
Solution: According to the question, a = 9cm and b= 12cm
θ = 90°
Area = ab sin (θ)
A = 9 × 12 sin (90)
A = 108sin 90
A = 108× 1 = 108 sq.cm.