## Area of Parallelogram

The parallelogram is a geometrical figure that is formed by the pair of parallel sides having opposite sides of equal length and the opposite angles of equal measure. The height and base of the parallelogram should be perpendicular to each other.

The area of a parallelogram is equal to the magnitude of cross-vector products for two adjacent sides. This is possible to create the area of a parallelogram by using any of its diagonals. The leaning rectangular box is a perfect example of the parallelogram.

### Proof Area of Parallelogram Forluma

According to the picture, Area of Parallelogram = Area of Triangle 1 + Area of Rectangle + Area of Triangle 2

=> Area of Parallelogram = \( \frac{1}{2} \times Height \times Base \) + \( Height \times Base \) + \( \frac{1}{2} \times Height \times Base \)

=> Area of Parallelogram = \( \frac{1}{2} \times h\times b_1\) + \( h\times b_3\) + \( \frac{1}{2} \times h\times b_2 \)

=> Area of Parallelogram = \( h ( \frac{1}{2} \times b_1 + b_3 + \frac{1}{2} \times b_2 \)

=> Area of Parallelogram = \( h { \frac{1}{2} (b_1 + b_2) } + b_3 \)

According to SAS, \( b_1 = b_2 \)

=> Area of Parallelogram = \( h { \frac{1}{2} (b_1 + b_1) } + b_3 \)

=> Area of Parallelogram = \( h { \frac{1}{2} \times 2 b_1 }+ b_3 \)

=> Area of Parallelogram = \( h ( b_1 + b_3 \)

According to picture \( b_1+ b_3 = Base \)

=> Area of Parallelogram = \( height \times Base \)

### Parallelogram Formula

A parallelogram is a four-sided polygon bounded by four infinite segments and it makes a closed figure that is referred to as the quadrilateral. Or we can say that the parallelogram is a special case of quadrilateral where opposite angles are equal and perpendicular to each other. With the help of a basic list of parallelogram formulas, you can calculate the area and the perimeter by putting the values and derive the final output. In the next section, we will discuss popular properties of the parallelogram for quick identification of shape.

Perimeter of Parallelogram \( = 2(a+b) \)

Diagonal of Parallelogram

=> \( p=\sqrt{a^{2}+b^{2}-2ab\cos (A)}=\sqrt{a^{2}+b^{2}+2ab\cos (B)} \)

=> \( q=\sqrt{a^{2}+b^{2}-2ab\cos (A)}=\sqrt{a^{2}+b^{2}-2ab\cos (B)} \)

=> \( p^{2}+q^{2}=2(a^{2}+b^{2}) \)

Where,

p,q are the diagonals

a,b are the parallel sides

### What is a Parallelogram?

In Euclidean Geometry, the parallelogram is the simplest form of a quadrilateral having two sides parallel to each other. The opposite sides of a parallelogram are equal in length and angles are also the same. The congruent sides are the direct consequence and it can be proved quickly with the help of equivalent formulations.

If only two sides are parallel then it is named as the Trapezoid and the three-dimensional counterpart is taken as the parallelepiped. A different type of quadrilateral on the basis of symmetry is defined as the given below –

**Rectangle**– This is a parallelogram with four equal angles and the opposite sides are also equal.**Rhombus**– This is a parallelogram with four sides of equal length.**Rhomboid**– This is a parallelogram whose opposite sides are parallel and adjacent or equal but its angles are not right-angled.**Square**–This is a parallelogram with four equal sides and angles of the same size.

### Properties of Parallelogram

- Two pairs of the opposite sides are equal in length and angles are of equal measure.
- The diagonals of a parallelogram bisect each other.
- The pair of opposite sides are equal and they are equal in length.
- The adjacent angles of the parallelogram are supplementary.
- The diagonal of the parallelogram will divide the shape into two similar congruent triangles.
- The shape has the rotational symmetry of the order two.
- Based on parallelogram law, the sum of the square of sides is equal to the sum of the square of the diagonals.
- The sum of the distances from any given point towards sides is equivalent to the location of the point.
- If there is point A in the plane of the quadrilateral then based on the property, every single line will divide the quadrilateral into two equal shapes.

Hence, a parallelogram could have all properties listed above if any of the statements become true then this is a parallelogram.

**Example1:** **If the base of a parallelogram is equal to 6cm and the height is 4cm, the find its area.**

Solution:

Base = 6 cm and height = 4 cm. Use the formula of Area of Parallelogram.

=> Area of Paralelogram = \( Height \times Base \) [Put the Value of Heigh and Base]

=> Area of Paralelogram = \( 4\times 6\)

=> Area of Paralelogram = \( 24 cm^2\)

**Example 2: The base of the parallelogram is thrice its height. If the area is 192 cm^{2}, find the base and height.**

Solution,

Given, Area of Parallelogram = 192 cm^{2}

Suppose Height =h and Base = 3h according to question.

Area of Parallelogram = \( Height \times base \)

=> \( 192 = h\times 3h\)

=> \( 192 = 3h^2\)

=> \( h^2 = \frac{192}{3}\)

=> \( h^2 = 64 \)

=> \( h^2 = 8^2 \)

=> \( h = 8 \)

Height =8 and Base = 3h= 24