LEARN THINGS THE EASY WAY

# What is a Parallelogram

By the end of this lesson, you will be able to

• define parallelogram.
• calculate the perimeter of parallelogram.
• evaluate the area of parallelogram.
• derive the formula for diagonal of parallelogram.
• interpret properties of parallelogram.
• demonstrate examples of parallelogram.

## Parallelogram

A convex quadrilateral with two pairs of parallel sides is a parallelogram.

A parallelogram is a type of quadrilateral.

Two parallel opposite sides for a parallelogram are of equal length.

The opposite angles are equal to each other.

The sum of two adjacent angles of a parallelogram adds up to 180°. Two diagonals of a parallelogram are different in length.

## Apps for calculating area and perimeter

sides and angle

side a:
side b:
angle θ:

Perimeter:

Area:

Select button and then drag the pink area of bottom right in figure and see the change.

## Parallelogram formula

The parallelogram formulas that are used generally given below:

## Perimeter of Parallelogram

The perimeter is the sum of lengths of all sides of a parallelogram.

Perimeter of a parallelogram can be evaluated depending on a pair of adjacent sides.

### Perimeter of Parallelogram Formula

Let a and b be the adjacent of a parallelogram.

The opposite sides of a parallelogram are of equal lengths.

If the perimeter of parallelogram is P,

P = (a + b + a + b) unit

or, P = (2a+2b) unit

∴ P = 2(a+b) unit

## Area of Parallelogram

Base and height of parallelogram

There are severals ways to finding out the area of a parallelogram depending upon the given conditions.

The area of a parallelogram is the base times the height.

### Area of Parallelogram Formula

Let b and h be the base and height respectively.

Area of parallelogram = (base × height) square unit i.e.

Area of parallelogram = (b × h) square unit

∴ Area of Parallelogram = bh

Base and height of para are shown in figure.

Now, One can have a question,why the area of a parallelogram is the base times height; and how? Let us derive the formula.

Base, height and diagonal of parallelogram is in figure.

Step 1: Draw the parallelogram ABCD.

Step 2: Draw a perpendicular AP from vertex A to the base BC i.e. AP⊥BC where AP = h and BC = b.

Step 3: Draw the diagonal AC.

Step 4: Diagonal of a parallelogram divides it two congruent triangles.
∴ ꕔABC ≅ ꕔACD.

Step 5: ▱ABCD = △ABC + △ACD

or, ▱ABCD = △ABC + △ABC [∵ △ABC ≅ △ACD]

or, ▱ABCD = 2.△ABC

or, ▱ABCD = 2× 12 BC.AP

or, ▱ABCD = BC.AP

∴ ▱ABCD = bh

∴ Area of parallelogram = (base × height) square unit (proved).

### Parallelogram Example

A parallelogram is shown in figure.

## Diagonal of Parallelogram Formula

Diagonal of a parallelogram means the length of a line segment connecting to opposite vertices.

Let a and b be the two adjacent sides and their inclined angle be θ.

Hence, the diagonal, d = √(a² + b²- 2ab cosθ).

## Properties of Parallelogram

• The four interior angles of a parallelogram add up to 3600.
• The adjacent angles are supplementary.
• Two diagonals of a parallelogram bisect to each other.
• One diagonal of a parallelogram divide it into two triangles of equal area.
• Both diagonals lie inside the parallelogram.
• The opposite sides are parallel.
• The opposite angles of parallelogram are equal to each other.
• The two triangles created by each of its diagonals are congruent.
• The opposite sides of a parallelogram are of equal lenghts.
• The sum of area of two squares formed by its two diagonals is equal to the sum of area of four squares formed by its four sides.