Area of Parallelogram

A parallelogram is a two-dimensional (2D) geometrical shape with sides that are parallel to one another. It is a type of polygon with four sides (also known as a quadrilateral), with each pair of parallel sides being the same length. A parallelogram's interior opposite angles are also similar. The sum of a parallelogram's adjacent angles equals 180 degrees.

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In the above-given figure, we can clearly see that ABCD is a parallelogram. 

Here, AB || CD and AD || BC. 

Also the sides, AB = CD and AD = BC

And angles, ∠A = ∠C & ∠B = ∠D

Also, here, ∠A & ∠D are supplementary angles as these interior angles lie on the same side of the transversal. In a similar way, ∠B & ∠C will be supplementary angles.

Hence,

∠A + ∠D = 180

∠B + ∠C = 180

Types of Parallelograms

A parallelogram can be classified into three types.

1. Rhombus: A rhombus is a parallelogram of equal sides.

A Rhombus is a parallelogram with equal-length sides on all four sides. A Rhombus is a quadrilateral of equilateral sides. We define an equilateral quadrilateral as a quadrilateral with equal sides. Since both pairs of opposite sides are parallel, every rhombus is a parallelogram. When all of the sides of a kite are the same length, it becomes a rhombus as the sides are equal and diagonals are perpendicular to each other. As a result, every rhombus is also a kite.

2. Rectangle: A parallelogram with all angles being right angles and diagonals being equal.

A rectangle is a special case of a parallelogram in which each interior angle is measured as 90 degrees. It's a quadrilateral with all equal angles. The term equiangular quadrilateral refers to the fact that all of the interior angles are of equal magnitude. Since the calculation of each angle is 90 degrees, the sum of its opposite angles is supplementary, indicating that it is a cyclic quadrilateral, with all of its vertices on the circumference of the circle.

3. Square: A square is a parallelogram with all equal sides and all 90-degree angles. A square's diagonals are also equal. A rhombus-like parallelogram with the properties of a rectangle. 

Properties of Parallelogram

If a quadrilateral has two parallel opposite sides, it is a special polygon known as a Parallelogram. A parallelogram has the following properties:

• Both sides will be parallel and congruent.

• The angles that are opposite each other are congruent.

• The consecutive angles will be supplementary.

• If any of the angles is a right angle, so all of the other angles are right angles as well.

• The two diagonals will bisect each other.

• The parallelogram is divided into two congruent triangles by each diagonal.

• The sum of the squares of all the sides of a parallelogram equals the sum of squares of its diagonals. It is also known as parallelogram law.

What is the Area of a Parallelogram?

A parallelogram's area is the region bounded by the parallelogram in a given two-dimensional space. To evaluate, a parallelogram is a special type of quadrilateral with four sides, two of which are parallel. A parallelogram has opposite sides that are equal in length and opposite angles that are equal in measurement. Since the rectangle and parallelogram have identical properties, the area of a rectangle equals the area of a parallelogram.

Area of Parallelogram Formula

A parallelogram's area is defined as the space enclosed by two pairs of parallel lines. Since a rectangle and a parallelogram have identical properties, the area of a parallelogram equals the area of a rectangle.

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Consider the parallelogram above, ABCD. The area of the parallelogram will be the space bounded by the sides AD, DC, CB, and AB.

The area of the parallelogram formula states that it is equal to the area of a rectangle.

Therefore, Area of a parallelogram = base x height

A = (b x h) Sq. units

Where b = the base of a parallelogram and,

h = The altitude or the height of a parallelogram.

The height or altitude is the perpendicular line (usually dotted) from the vertex of a parallelogram to any of the bases.

Example 1: Find the area of a parallelogram whose base is 9 centimeters and height is 7 centimeters.

Solution: We know that Area = (b x h) Sq. units

Where b = the base of a parallelogram and, h = The altitude or the height of a parallelogram.

Therefore,  A = (9 x 7) = 63 cm2

How to Find the Area of a Parallelogram without a Height?

If we don’t know the parallelogram’s height, then we can use the trigonometry concept to find the area of the parallelogram.

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Area = ab sine (α) =ab sine (β)

Here, a and b are the length of parallel sides. 

And either β or α is the angle between the sides of the parallelogram as shown in the fig above. 

Example 2: Find the area of a parallelogram if its two parallel sides are 80 cm and 40 cm and the angle between them is 56 degrees.

Solution: It is given that, a = 80 cm & b = 40 cm.

The angle between the lines a and b is also given as, α = 56 degrees.

We know that Area = ab sine (α) =ab sine (β)

Let’s substitute the values of a, b & α.

We get,

A = 80 × 40 sine (56)

A = 3,200 sine 56

A = 2,652.9 sq.cm.

Area of Parallelogram Using Diagonals

We can also find the area of any parallelogram using diagonal lengths of Parallelogram. 

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As we already know that there are 2 diagonals for any parallelogram, which intersect each other. 

Here, d1 and d2 are the diagonals of parallelogram ABCD, then the area of the parallelogram will be given as,

A = ½ × d1 × d2 sine (β) = ½ × d1 × d2 sine (α)

Where β or α is the angle of intersection of the diagonals d1 and d2.

Perimeter of Parallelogram

The perimeter of any shape is the total distance covered around the shape or its total length. Similarly, the perimeter of a parallelogram is the total distance of the parallelogram's edges. To measure the perimeter value, we must first know the length and breadth of the object. The parallelogram has opposite sides that are the same length.

Therefore, the formula to calculate the perimeter can be written as:

Perimeter = 2 (a+b) units 

Where a & b are sides of the parallelogram. 

Conclusion

A parallelogram's area is the region bounded by the parallelogram in a given two-dimensional space. To examine, a parallelogram is a special type of quadrilateral with four sides, two of which are parallel. A parallelogram has opposite sides that are equal in length and opposite angles that are similar in magnitude. Since the rectangle and parallelogram have identical properties, the area of a rectangle equals the area of a parallelogram. A parallelogram has opposite angles that are congruent, and consecutive angles that are supplementary. We've also seen other shapes with similar properties to parallelograms, such as squares, rectangles, and rhombuses.