A parallelogram is a four-sided plane figure. It has two pairs of parallel sides. It is similar to a rectangle, but the opposite sides are not necessarily equal in length. A parallelogram has four angles, two of which are acute and two of which are obtuse. The acute angles are opposite to each other, as are the obtuse angles.

## What is the Area of Parallelogram?

The area of a parallelogram is the amount of surface that is enclosed by its four sides.

**Area of Parallelogram = base x height**

where,

- the base is one of the parallel sides of the parallelogram,
- and the height is the perpendicular distance between the two parallel sides.

In other words, the height is the shortest distance between the two parallel sides.

## Perimeter of a Parallelogram

The perimeter of a parallelogram can be found by adding the lengths of all four sides. If the parallelogram has sides of length a and b, and one of its angles is θ, then the perimeter can be calculated using the formula:

**Perimeter = 2(a+b)/cos(θ)**

## How to Calculate the Area of Parallelogram?

To calculate the area of a parallelogram, follow these steps:

- Measure the length of one of the parallel sides of the parallelogram. This is called the base.
- Measure the distance between the base and the opposite side of the parallelogram. This is called the height.
- Multiply the length of the base by the height to get the area of the parallelogram.
- Make sure to include units of measurement for your answer, such as square centimeters or square meters, depending on the unit of measurement used for the base and height.

### Area of Parallelogram Without Height

If the height of the parallelogram is not given, you can use another formula to find the area using the lengths of the sides and the included angle.

**Area = base x side x sin(angle)**

where base is the length of one of the sides of the parallelogram, side is the length of the adjacent side, and angle is the included angle between those two sides.

**For example, if you know the lengths of the sides of a parallelogram are 5 cm and 8 cm, and the included angle is 60 degrees, then find the area.**

Area = 5 cm x 8 cm x sin(60 degrees) = 20 cm^2 x √3/2 = 10√3 cm^2

so, the area of the parallelogram is 10√3 square centimeters.

### Area of Parallelogram Using Diagonals

If you know the lengths of the diagonals of a parallelogram, you can use them to find the area of the parallelogram using the following formula:

**Area = 1/2 x d1 x d2**

where d1 and d2 are the lengths of the diagonals of the parallelogram.

**For example, a parallelogram with diagonals of length 6 cm and 8 cm. Find the area of the parallelogram using these diagonals.**

Area = 1/2 x d1 x d2 = 1/2 x 6 cm x 8 cm = 24 cm^2

Therefore, the area of the parallelogram is 24 square centimeters.

## Area of Parallelogram in Vector Form

The area of a parallelogram can be expressed in terms of the vectors representing its adjacent sides. If the parallelogram has sides given by vectors u and v, the area of the parallelogram can be found using the following formula:

**Area = |u x v|**

where x represents the cross product of the vectors u and v, and |u x v| represents the magnitude of the resulting vector.

**For example, a parallelogram with adjacent sides represented by the vectors u = [3,2] and v = [4,1]. Fnd the area of the parallelogram.**

- Take the cross product of u and v:u x v = (3i + 2j) x (4i + j) = 3(4j) – 2(4i) – 2(j) = 4j – 8i – 2j = -8i + 2j
- Find the magnitude of the resulting vector:|u x v| = sqrt((-8)^2 + 2^2) = sqrt(68) = 2sqrt(17)

## Solved Examples on Area of Parallelogram

**Example 1: Find the area of a parallelogram with base 10 cm and height 8 cm.**

**Solution:** The area of a parallelogram is given by the formula: Area = base x height

Substituting the given values, we get: Area = 10 cm x 8 cm Area = 80 cm^{2}

Therefore, the area of the parallelogram is 80 square centimeters.

**Example 2: Find the area of a parallelogram with sides of length 5 cm and 8 cm, and the included angle of 60 degrees.**

**Solution:** The area of a parallelogram is given by the formula: Area = base x side x sin(angle)

Substituting the given values, we get: Area = 5 cm x 8 cm x sin(60 degrees) Area = 40 cm^{2}x √3/2 Area = 20√3 cm^{2}

Therefore, the area of the parallelogram is 20√3 square centimeters.

**Example 3: Find the area of a parallelogram with diagonals of length 10 cm and 12 cm.**

Solution: The area of a parallelogram can be found using the formula: Area = 1/2 x d1 x d2

Substituting the given values, we get: Area = 1/2 x 10 cm x 12 cm Area = 60 cm^{2}

Therefore, the area of the parallelogram is 60 square centimeters.

**Example 4: Find the area of a parallelogram with adjacent sides given by the vectors u = [2,4] and v = [3,-1].**

Solution: The area of a parallelogram can be found using the formula: Area = |u x v|

Taking the cross product of the vectors u and v, we get: u x v = (2i + 4j) x (3i – j) = 2(3j) – 4(3i) + 4(i) = -12i + 6j

The magnitude of the resulting vector is: |u x v| = sqrt((-12)^2 + 6^2) = sqrt(180) = 6sqrt(5)

Therefore, the area of the parallelogram is 6sqrt(5) square units

## Frequently Asked Questions

**Q: What is a parallelogram?**

A: A parallelogram is a four-sided figure with opposite sides that are parallel to each other. Its opposite sides are also congruent (of equal length) and its opposite angles are also congruent (of equal measure).

**Q: What is the formula for the area of a parallelogram?**

A: The formula for the area of a parallelogram is given by: Area = base x height, where the base is the length of one of the parallel sides and the height is the perpendicular distance between the base and the opposite parallel side.

**Q: Can you find the area of a parallelogram without knowing the height?**

A: Yes, there are different methods to find the area of a parallelogram even without knowing the height. One method is to use the lengths of the diagonals, another is to use the lengths of the adjacent sides and an included angle, and a third method is to use vectors to represent the adjacent sides.

**Q: Can the area of a parallelogram be negative?**

A: No