A **parallelogram** is a two-dimensional geometrical shape, whose sides are parallel with each other. It is made up of four sides, where the pair of parallel sides are equal in length. Also, the opposite angles of a parallelogram are equal to each other. The area of parallelogram depends on the base and height of it.

In geometry, you must have learned about many 2D shapes and sizes such as circle, square, rectangle, rhombus, etc. All of these shapes have a different set of properties. Also, the area and perimeter formulas of these shapes vary with each other, used to solve many problems. Let us learn here the definition, formulas and properties of a parallelogram.

**Table of contents:**

## Parallelogram Definition

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length and the opposite angles are equal in measure.Â

In the figure above, you can see, ABCD is a parallelogram, where AB//CD and AD//BC.Â

Also, AB = CD and AD = BC

And,Â âˆ A =Â âˆ C &Â âˆ B =Â âˆ D

**Also, read:**

## Parallelogram Formula

The formula for area and parameter of a parallelogram covered here in this section. Students can use these formulas and solve problems based on them.

### Area of Parallelogram

Area of a parallelogram is the region occupied by it in a two-dimensional plane. Below is the formula to find the parallelogram area:

Area = Base Ã— Height

In the above figure, //^{gram}ABCD,Â Area is given by;

Area = a b sin A = b a sin B |

where a is the slant length of the side of //^{gram}ABCD and b is the base.

**Check here:**Â Area of a Parallelogram Formula

### Perimeter of Parallelogram

The perimeter of any shape is the total distance of the covered around the shape or its total length of any shape. Similarly, theÂ **perimeter of a parallelogram**Â is the total distance of the boundaries of the parallelogram. To calculate the perimeter value we have to know the values of its length and breadth. The parallelogram has its opposite sides equal in length. Therefore, the formula of the perimeter could be written as;

Perimeter = 2 (a+b)Â |

Where a and b are the length of the equal sides of the parallelogram.

## Properties of Parallelogram

If a quadrilateral has a pair of parallel opposite sides, then itâ€™s a special polygon called Parallelogram. The properties of a parallelogram are as follows:

- The opposite sides are congruent.
- The opposite angles are congruent.
- The consecutive angles are supplementary.
- If anyone of the angles is a right angle, then all the other angles will be right.
- The two diagonals bisect each other.
- Each diagonal bisects the parallelogram into two congruent triangles.
- The diagonals separate it into congruent.

## Types of Parallelogram

There are mainly four types of Parallelogram depending on various factors. The factors which distinguish between all of these different types of parallelogram are angles, sides etc.

- In a parallelogram, say PQRSÂ
- If PQ = QR = RS = SP are the equal sides, then itâ€™s a rhombus. All the properties are the same for rhombus as for parallelogram.

- Other two special types of a parallelogram are:

- Rectangle
- Square

### Is Square a Parallelogram?

Square could be considered as a parallelogram since the opposite sides are parallel to each other and the diagonals of the square bisect each other.

### Is Rectangle a Parallelogram?

Yes, a rectangle is also a parallelogram, because satisfies the conditions or meet the properties of parallelogram such as the opposite sides are parallel and diagonals intersect at 90 degrees.

## Parallelogram Theorems

**Theorem 1:**Â **Parallelograms on the same base and between the same parallel sides are equal in area.**

**Proof:**Â Two parallelograms ABCD and ABEF, on the same base DC and between the same parallel line AB and FC.

To prove that area (ABCD) = area (ABEF).

Proof:

Parallelogram ABCD and rectangle ABML are on the same base and between the same parallels AB and LC.

area of parallelogram ABCD = area of parallelogram ABML

We know that area of a rectangle = length x breadth

Therefore, area of parallelogram ABCD = AB x AL

Hence, the area of a parallelogram is the product of any base of it and the corresponding altitude.

In âˆ†ADF and âˆ†BCE,

AD=BC (âˆ´ABCD is a parallelogram âˆ´ AD=BC)

AF=BE (âˆ´ABEF is a parallelogram âˆ´AF=BE)

âˆ ADF=âˆ BCE (Corresponding Angles)

âˆ AFD=âˆ BEC (Corresponding Angles)

âˆ DAF =âˆ CBE (Angle Sum Property)

âˆ†ADE â‰… âˆ†BCF (From SAS-rule)

Area(ADF) = Area(BCE) (By congruence area axiom)

Area(ABCD)=Area(ABED) + Area(BCE)

Area(ABCD)=Area(ABED)+Area(ADF)

Area(ABCD)=Area(ABEF)

Hence, the area of parallelograms on the same base and between the same parallel sides is equal.

**Corollary**: **A parallelogram and a rectangle on the same base and between the same parallels are equal in area.**

**Proof:**Â Since a rectangle is also a parallelogram so, the result is a direct consequence of the above theorem.

**Theorem:**Â **The area of a parallelogram is the product of its base and the corresponding altitude.**

**Given:**Â In a parallelogram ABCD, AB is the base.

**To prove**Â that Area(||^{gm}ABCD) = ABÃ—AL

**Construction:**Â Complete the rectangle ALMB by Drawing BM perpendicular to CD.

## Examples of Parallelogram

Example- Find the area of a parallelogram whose base is 5 cm and height is 8 cm.
We know, Area = Base x Height Area = 5Â Ã— 8Â Area = Â 40 Sq.cm
The angle opposite to the side b comes out to be 180 – 65 = 115Â° We use the law of cosines to calculate the base of the parallelogram – bÂ² =Â 5Â² + 11Â² – 2(11)(5)cos(115Â°) bÂ² =Â 25 + 121 – 110(-.422) bÂ² = 192.48 b = 13.87 cm. After finding the base we need to calculate the height of the given parallelogram. To find the height we have to calculate the value of Î¸, so we use sine law 5/sin(Î¸) = b/sin(115) Now we extend the base and draw in the height of the figure and denote it as ‘h’. The right-angled triangle (marked with red line) has the Hypotenuse to be 22 cm and Perpendicular to be h. So sin Î¸ = h/22 h = 7.184 cm Area = base Ã—Â height A = 13.87 Ã— 7.184 A = 99.645 sq.cm |

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