In geometry a parallelogram is a simple quadrilateral with opposite sides being parallel and equal in length. Opposite angles of parallelogram will be of equal measure. Area of a parallelogram can be found by multiplying base length with the corresponding altitude.

As the opposite sides of a parallelogram are parallel they will never intersect. In the below diagram angle a and angle b add up to 180$^{\circ}$.

## Area of a Parallelogram Formula

Area of a parallelogram is the number of square units it takes to completely fill a parallelogram.
Area of a parallelogram is given by the formula:

Area = b $\times$ h
where b : base of the parallelogram (length of any base)
h : height of the parallelogram (altitude)

The height and base of the parallelogram will be perpendicular to each other always. If 'height' of a parallelogram is not known then there is an alternative formula which can be used and is given by

Area = ab sin $\theta$
Where a and b are the known sides in a parallelogram and $\theta$ is the known angle.

## Surface Area of a Parallelogram

Surface area is the area of a given surface. It is the total area of the faces and curved faces of a solid figure and is denoted by "S". In general surface area is the sum of all the areas of all the shapes that cover the surface of the object.

Surface area of a parallelogram is same as the formula for area of a parallelogram. Therefore the formula for surface area of a parallelogram is
Surface area = b $\times$ h
where b : base and h : height.

## Find The Area of a Parallelogram with Vertices

Area of a parallelogram with vertices can be found by using vector cross product or by finding the sum of the areas of two triangles.

Area of a parallelogram with vertices by using vector cross product

The formula is, Area = ab sin $\theta$
Where a and b are the known sides in a parallelogram and $\theta$ is the known angle.

Area of a parallelogram with vertices by using sum of the areas of two triangles

For a parallelogram ABCD draw a diagonal AC. You get two triangles ABC and ACD. Find the area of these two triangles and add them.
The formula to find the area of the triangle ABC is given by:

Area of $\triangle$ABC = $\frac{1}{2}$ [$x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}$)]
Similarly, find the area of the triangle ACD.

Therefore, Area of parallelogram ABCD = (Area of triangle ABC + Area of triangle ACD)

### Solved Example

Question: Find the area of the parallelogram with vertices A (1, -2), B(2, -2), C(5, 3), D(-4, 1)
Solution:

For a parallelogram ABCD, firstly find the area of two triangles ABC and ACD.
To find the area of triangle ABC

Let   ($x_{1}, y_{1}$) =   A (1, -2)
($x_{2}, y_{2}$) = B (2, -2)
($x_{3}, y_{3}$) = C (5, 3)

Area of $\triangle$ ABC = $\frac{1}{2}$ [$x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}$)]

= $\frac{1}{2}$ [1(- 2 - 3) + 2 (3 + 2 ) + 5 ( - 2 + 2)]

Area of $\triangle$ ABC = $\frac{5}{2}$

The area of triangle ACD

Let   ($x_{1}, y_{1}$) = A (1, -2)
Let   ($x_{2}, y_{2}$) = C (5 , 3)
Let   ($x_{3}, y_{3}$) = D (-4, 1)

Area of $\triangle$ ACD = $\frac{1}{2}$ [$x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}$)]

= $\frac{1}{2}$ [ (1 (3 - 1) + 5 (1 + 2)  - 4 ( -2 - 3)

Area of $\triangle$ ACD = $\frac{37}{2}$

Therefore Area of parallelogram ABCD = (Area of triangle ABC + Area of triangle ACD)

Area of parallelogram ABCD = $\frac{5}{2}$ + $\frac{37}{2}$

Area of parallelogram ABCD = 21 square units.

## How to Find The Area of a Parallelogram

### Solved Examples

Question 1: Find the area of a parallelogram with a base of 14 cm and a height of 12 cm.
Solution:

The formula to find the area of a parallelogram is

Area = b $\times$ h

Area = (14 cm) (12cm)

Area = 168 cm$^{2}$

Therefore the area of a parallelogram is 168 cm$^{2}$.

Question 2: The area of a parallelogram is 13 m$^{2}$ and the base of the parallelogram is 7 m. Find its height.
Solution:

The formula to find the area of a parallelogram is

Area = b $\times$ h

13 = 7h

h = $\frac{13}{7}$

h = 1.86

Therefore the height of the parallelogram is 1.86 m.