Bhargav

· 14 Dec 2021

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How to find the area of the parallelogram

A parallelogram's area is the area that a parallelogram occupies in a two-dimensional plane. A parallelogram is a two-dimensional figure with four sides in geometry. It is a quadrilateral, with equal and parallel opposite sides. A parallelogram's area is defined as the space enclosed by its four sides. The parallelogram's area is equal to the product of its length and height.

The total of a quadrilateral's internal angles equals 360 degrees. A parallelogram comprises two pairs of parallel sides of equal length. Because it is a two-dimensional figure, it contains an area and a perimeter. Let us now analyze the area of a parallelogram in detail, including its formula, derivations, and further solved problems.

What is the Parallelogram's Area?

The area of a parallelogram is the territory included within it in a two-dimensional space. A parallelogram is a special sort of quadrilateral with four sides, two of which are parallel to refresh your memory. Parallelograms have opposite sides that are equal in length and opposite angles that are equal in size. Due to the identical features of the rectangle and the parallelogram, the rectangle area is equal to the area of a parallelogram.

The formula for Parallelogram Area

To determine the parallelogram's area, multiply the perpendicular base by the height. It should be observed that the parallelogram's base and height are perpendicular to one another, but the parallelogram's lateral side is not perpendicular to the base. As a result, a dotted line is used to denote the height.

How to Calculate the Parallelogram's Area?

The area of a parallelogram can be computed using the base and height of the parallelogram. Apart from that, the area of a parallelogram may be calculated if the lengths of the parallel sides and any of their intersecting angles are known or if the lengths of the parallel sides and any of their intersecting angles are known. As a result, there are three methods for calculating the area of a parallelogram:

•    When the parallelogram's base and height are specified
•    When the height is not specified
•    When given diagonals

Parallelogram Area Calculated Using Sides

If a and b are the parallel sides of a parallelogram and h is its height, then the formula for its area is as follows:

A = b h [sq. ft.] Area = Base Height A = b h [sq ft] .unit]
Example: If the base of a parallelogram is 5 cm and the height is 3 cm, determine the area of the parallelogram.
Solution: Given a base length of 5 cm and a height of 3 cm
Area = 5 3 = 15 sq.cm

Parallelogram's Area without Height

If we do not know the height of the parallelogram, we can apply the trigonometry idea to determine its area.
Area = sin ab (x)
A and b denote the lengths of parallel sides, and x denotes the angle formed by the parallelogram's sides.

Example: The angle formed by any two parallelogram sides is 90 degrees. Determine the area if the lengths of the two parallel sides are 3 cm and 4 cm, respectively.
Solution: Assume that a = 3 cm and b = 4 cm and that x = 90 degrees.
Area = sin ab (x)
A=3 4 sin (90)
12 sin 90 = A
A = 12 1 Equals 12 square centimeters.

Parallelogram Area Calculated Using Diagonals

The area of a parallelogram can also be determined using the diagonal lengths of its parallelograms. As is well known, a parallelogram has two intersecting diagonals. If the diagonals connect at an angle y, then the parallelogram's area is given by:
12 d1 d2 sin (y)

Parallelogram Area in Vector Form

If the sides of a parallelogram are supplied in vector form, the parallelogram's area can be determined using the parallelogram's diagonals. Assume that vectors 'a' and 'b' represent the two sides of a parallelogram and that the resulting vector is the parallelogram's diagonal.
The area of a parallelogram in vector form equals the modulus of the cross product of vectors a and b.

A = | a b

In vector form, we must find the area of a parallelogram about two diagonals, say d1 and d2.
Thus, a + b = d1 b + (-a) = d2 or b – a = d2
As a result, d1 d2 = (a + b). (b – a) = a (b – a) + b (b – a) = a b – an a + b b – b a = a b – 0 + 0 – b a
Given that a b = – b a
As a result, d1 d2 = a b + a b = 2 (a b).
A b = half (d1 d2)

Area of a Parallelogram Practice Questions

1.    Determine the area of a parallelogram with an 8 cm base and a 4 cm height.
2.    Determine the area of a parallelogram with a 7-inch base and a 9-inch height.
3.    The parallelogram's base is three times its height. If the area of a rectangle is 147 square units.
4.    A parallelogram has sides of ten and eight meters. If the distance between the shortest sides is 5m, determine the distance between the parallelogram's longest sides.




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