Parallelogram is derived from the Greek word “parallelogrammon” meaning “limited by parallel lines”. So, a parallelogram is a quadrilateral surrounded by parallel lines. Parallelogram is a basic figure in geometry and is a quadrilateral. It is a quadrilateral in which opposite sides are parallel and equal. Parallelogram also has opposite pairs of angles as equal.

In this article, we will learn more about the properties of a parallelogram, the area, and perimeter of parallelograms, and their examples, in detail.

## What is a Parallelogram?

A parallelogram is one of the special types of quadrilateral. The angles between adjacent sides of a parallelogram can vary, but to be a parallelogram, the opposite sides must be parallel. A quadrilateral is a parallelogram if the opposite sides are parallel and equal. Therefore, a parallelogram is defined as a quadrilateral in which two pairs of opposite sides are parallel and equal. Below is the diagram of a parallelogram ABCD having adjacent sides ‘a’ and ‘b’ and height ‘h’.

## Real-Life Examples of a Parallelogram

We came across various things in our daily life which resembles a parallelogram. Such as the computer screen, books, buildings, and tiles all are considered to be in a parallelogram shape.

The parallelogram is the most common shape which we encounter daily. Rectangle and square both can be considered a parallelogram and are easily seen in our daily life.

## Shape of Parellelogram

A parallelogram is a two-dimensional closed shape with four sides, i.e. it is a quadrilateral. In parallelogram opposite sides are always parallel and equal. Opposite pairs of angles are also equal in parallelograms. In a parallelogram, the interior angle can and can not be a right angle.

Thus, all rectangles and squares can be considered parallelograms but the opposite is not true, i.e. all parallelograms are not considered to be squares or rectangles.

## Special Parallelograms

Some of the special parallelograms which are widely used in geometry are,

Rectangle**:** It is a special case of the parallelogram in which all the interior angles are equal and their value is equal to 90°.

Square**: **It is a parallelogram in which all the sides and all the angles are equal and the measure of each interior angle is 90°.

Rhombus**:** Rhombus is also a parallelogram in which all the sides are equal but all the angles are not equal even though opposite pairs of angles are equal.

## Angles of Parallelogram

Parallelogram is a quadrilateral i.e. a polygon with four sides and four angles and the opposite pair of angles are equal in the parallelogram. i.e. in a parallelogram ABCD ∠A is equal to ∠C and ∠B is equal to ∠D.

The sum of all the angles in the quadrilateral is 360°. As a parallelogram is a quadrilateral so sum of all the angles of the parallelogram ABCD equals 360°. Now,

∠A + ∠B + ∠C + ∠D = 360°

in parallelogram ∠A = ∠C and ∠B = ∠D

Thus,

∠A + ∠B + ∠A + ∠B = 360°

2(∠A + ∠B) = 360°

∠A + ∠B = 180°

Similarly, ∠C + ∠D = 180°

Thus, Adjacent angles are supplementry in a parallelogram.

## Properties of Parallelogram

There are some special properties of quadrilateral, if applied, make it a parallelogram. Let’s take a look at the properties of parallelograms,

- The opposite sides of a parallelogram are parallel and equal.
- The opposite angles of a parallelogram are congruent.
- If said, one of the angles of a parallelogram is 90°. Then all the angles are 90°, and it becomes a rectangle.
- The diagonals of the parallelogram bisect each other.
- The consecutive angles of a parallelogram are supplementary.

## Types of Parallelogram

Parallelograms can be classified into several types based on their properties. Based on the properties of a parallelogram and on the sides and angles, the parallelogram is classified into three types,

### Rectangle

A rectangle is a parallelogram with two pairs of equal, parallel opposite sides and four right angles. Observe the rectangle ABCD and associate it with the following properties,

- Two pairs of parallel sides. Here AB || DC and AD || BC
- Four right angles ∠A = ∠B = ∠C = ∠D = 90°.
- The opposite sides are the same length, where AB = DC and AD = BC.
- Two equal diagonals where AC = BD.
- Diagonals that bisect each other.

### Square

A parallelogram with four equal sides and four equal angles is called a square. Observe the square ACDB and associate it with the following properties,

- Four equal sides are AB = BC = CD = DA.
- Right angles are ∠A = ∠B = ∠C = ∠D = 90°.
- There are two pairs of parallel sides. Here AB || DC and AD || BC.
- Two identical diagonals where AD = BC.
- Diagonals are perpendicular and bisect each other; AD is perpendicular to BC.

### Rhombus

A parallelogram with four equal sides and equal opposite angles is called a rhombus. Consider the diamond ABCD and assign it the following attributes,

- In the given figure, the four equal sides are AB = CD = BC = AD.
- The two pairs of parallel sides are AB || CD and BC || AD.
- The equal opposite angles are ∠A = ∠B and ∠C = ∠D.
- Here, the diagonals (AC and BD) are perpendicular to each other and bisect at right angles.

All 2D shapes have two basic formulas for area and perimeter. Let’s discuss these two parallelogram formulas in this section.

## Parallelogram Formulas

Parallelogram is a basic 2-dimensional figure which is widely used in mathematics it has various formulas some of them are of Areas and Perimeter

- Area of Parallelogram
- Perimeter of Parallelogram

Let us discuss these two formulas in detail

## Area of Parallelogram

The area of a parallelogram is the space between the four sides of the parallelogram. It can be calculated by knowing the length of the base and the height of the parallelogram and measuring it in square units such as cm^{2}, m^{2}, or inch^{2}. Note the following parallelogram representing the base and height.

Consider a parallelogram ABCD with a base (b) and a height (h). The area of a parallelogram is calculated by the formula:

Parallelogram area = base (b) × height (h)

### Area of Parallelogram without Height

When the height of the parallelogram is not known, the area of the parallelogram can still be found, provided the angle is known to us. The formula for the area of a parallelogram without height is given as:

Parallelogram area = ab Sinθ

Where a and b are the sides of the parallelogram and θ is the angle between them.

## Perimeter of Parallelogram

The perimeter of a parallelogram is the length of its contour, so it is equal to the sum of all sides. In a parallelogram, the opposite sides are equal. Let’s say the sides are a and b. Therefore, the perimeter (P) of a parallelogram with edges is in units of P = 2 (a + b).

Perimeter of Parallelogram = 2 (a + b)

## Parallelogram Theorem

**Theorem:** Parallelograms on the same base and between the same parallels have equal area.

**Proof:** Let’s assume two parallelograms ABCD and ABEF with the same base DC and between the same parallel lines AB and FC

**To Prove:** Area of parallelogram ABCD = Area of parallelogram ABEF

In the figure given below, the two parallelograms, ABCD and ABEF, lie between the same parallel lines and have the same base. Area ABDE is common between both parallelograms. Taking a closer look at the two triangles, △BCD and △AEF might be congruent.

BC = AE (Opposite sides of a parallelogram),

∠BCD = ∠AEF (These are corresponding angles because BC || AE and CE are the transversal).

∠BDC = ∠AFE (These are corresponding angles because BD || AF and FD are the transversals).

Thus, by the ASA criterion of congruent triangles. These two triangles are congruent, and they must have equal areas.

area(BCD) = area(AEF)

area(BCD) + area(ABDE) = area(AEF) + area(ABDE)

area(ABCD) = area(ABEF)

Hence, parallelograms lying between the same parallel lines and having a common base have equal areas.

## Difference Between Rectangle and Parallelogram

Rectangle and parallelogram are both quadrilaterals, and the rectangle is a parallelogram as it has all the properties of a parallelogram and more. However, a parallelogram is not always a rectangle. Below are the differences in the properties of a rectangle and a parallelogram.

Properties | Parallelogram | Rectangle |
---|---|---|

Sides | The opposite sides of a parallelogram are equal. | The opposite sides of a rectangle are equal. |

Diagonals | The diagonals of a parallelogram bisect each other, but the diagonals are not equal. | The diagonals of a rectangle bisect each other, and the diagonals are equal to each other as well. |

Angles | The opposite angles of a parallelogram are equal, and the adjacent angles are supplementary. | All the angles of a rectangle are equal to each other and equal to 90°. |

**Do Check,**

- Area of a Triangle
- Area of a Square
- Area of Rectangle

## Solved Examples on Parallelogram

**Example 1: Find the length of the other side of a parallelogram with a base of 12 cm and a perimeter of 60 cm.**

**Solution:**

Given perimeter of a parallelogram = 60cm.

Base length of given parallelogram = 12 cm.

P = 2 (a + b) units

Where b = 12cm and P = 40cm.

60 = 2 (a + 12)

60 = 2a + 24

2a = 60-24

2a = 36

a = 18cm

Therefore, the length of the other side of the parallelogram is 18 cm.

**Example 2: Find the perimeter of a parallelogram with the base and side lengths of 15cm and 5cm, respectively.**

**Solution:**

Base length of given parallelogram = 15 cm

Side length of given parallelogram = 5 cm

Perimeter of a parallelogram is given by,

P = 2(a + b) units.

Putting the values, we get

P = 2(15 + 5)

P = 2(20)

P = 40 cm

Therefore, the perimeter of a parallelogram will be 40 cm.

**Example 3: The angle between two sides of a parallelogram is 90°. If the lengths of two parallel sides are 5 cm and 4 cm, respectively, find the area.**

**Solution:**

If one angle of the parallelogram is 90°. Then, the rest of the angles are also 90°. Therefore, the parallelogram becomes a rectangle. The area of the rectangle is length times breadth.

Area of parallelogram = 5 × 4

Area of parallelogram = 20cm

^{2}(Video) Properties of Parallelograms (with practice)

**Example 4: Find the area of a parallelogram when the diagonals are given as 8 cm, and 10 cm, the angle between the diagonals is 60°.**

**Solution:**

In order to find the area of the parallelogram, the base and height should be known, lets’s first find the base of the parallelogram, applying the law of cosines,

b

^{2}= 4^{2}+ 5^{2}– 2(5)(4)cos(120°)b

^{2 }= 16 + 25 – 40(0.8)b

^{2}= 9b = 3cm

Finding the height of the parallelogram,

4/sinθ = b/sin120

4/sinθ = 3/-0.58

sinθ = -0.773

θ = 50°

Now, to find the height,

Sinθ = h/10

0.76 = h/10

h = 7.6cm

Area of the parallelogram = 1/2 × 3 × 7.6

= 11.4 cm^{2}

**Example 5: Prove that a parallelogram circumscribing a circle is a rhombus.**

**Solution:**

Given:

- Parallelogram ABCD
- Circle PQRS
To prove: ABCD is a rhombus.

Proof:

We know that the tangents drawn from an exterior point to a circle are equal to each other. Therefore:

AP = AS ⇢ (1)

BP = BQ ⇢ (2)

DS = DR ⇢ (3)

CR = CQ ⇢ (4)

(Video) What is a Parallelogram? | Special Cases of Parallelogram | Don't MemoriseAdding the LHS and RHS of equations 1, 2, 3, and 4:

AP + BP + DS + CR = AS + BQ + DR + CQ

AB + DR + CR = AS + DS + BC

AB + CD = AD + BC

Since the opposite angles of a parallelogram are equal:

2AB = 2BC

AB = BC, and similarly, CD = AD.

Therefore: AB = CD = BC = AD.

Since all the sides are equal, ABCD is a rhombus.

## FAQs on Parallelogram

### Q1: What is a parallelogram?

**Answer:**

A parallelogram is a quadrilateral with opposite pair of lines as parallel and equal.

### Q2: What is the Area of a Parallelogram?

**Answer:**

The space occupied inside the boundary of the triangle is termed the area of the parallelogram. It can be calculated using the formula,

Area of Parallelogram = Base (b) × Height (h)

### Q3: What is the Perimeter of a Parallelogram?

**Answer:**

The length of all the boundaries of the triangle is termed the perimeter of the parallelogram. It can be calculated using the formula,

Perimeter of Parallelogram = 2(l + b)where,

lis the length of parallelogrambis the base of parallelogram

### Q4: Does a parallelogram have equal diagonals.

**Answer:**

No, the diagonals of a parallelogram are not equal. However, the diagonals of a parallelogram bisect each other.

### Q5: How many lines of symmetry parallelogram have?

**Answer:**

In general, a parallelogram has none or 0 lines of symmetry. But in special cases of parallelogram line of symmetry is present. Lines of symmetry of special parallelograms are,

Square 4 Rhombus 2 Rectangle 2

### Q6: Does a parallelogram have equal sides?

**Answer:**

Yes, all parallelograms have equal pairs of opposite sides but not all sides are equal in a parallelogram.

**Q7: Is a rhombus a parallelogram?**

**Answer:**

Yes, a rhombus is a parallelogram. A rhombus has all the properties of the parallelogram and more.

**Q8: How is a parallelogram different from a quadrilateral?**

**Answer:**

All parallelograms are quadrilaterals, but not all quadrilaterals are necessarily parallelograms. For example, a trapezoid is a quadrilateral, not a parallelogram. For a quadrilateral to be a parallelogram, all opposite sides must be parallel and equal.

**Q9: Is a square a parallelogram?**

**Answer:**

Yes, a square is a parallelogram as it has all the properties of a parallelogram and more since a square has some extra properties (For example, all angles are right angles, etc.), all parallelograms are not squares.

**Q10: Is a rectangle a parallelogram?**

**Answer:**

Yes, a rectangle is a parallelogram. Rectangle is a parallelogram as it has all the properties of a parallelogram and more. However, a parallelogram is not always a rectangle.

## FAQs

### Parallelogram - Definition, Types, Formulas, Examples, and FAQs? ›

**A parallelogram is a quadrilateral (a polygon with four sides) in which the opposite angles are equal and the diagonals bisect each other**. Bisecting a parallelogram along a diagonal divides the figure into two congruent triangles. Rectangles, rhombuses, and squares are all special types of parallelograms.

**What are the 4 types of parallelograms? ›**

Parallelograms are shapes that have four sides with two pairs of sides that are parallel. The four shapes that meet the requirements of a parallelogram are **square, rectangle, rhombus, and rhomboid**. A rhombus looks like a slanted square, and a rhomboid looks like a slanted rectangle.

**What is parallelogram examples and definitions? ›**

In a parallelogram, the opposite sides are parallel and equal in length. A few examples of a parallelogram are **rhombus, rectangle, and square**.

**What are the parallelogram related formulas? ›**

All Formulas to Calculate Area of a Parallelogram | |
---|---|

Using Base and Height | A = b × h |

Using Trigonometry | A = ab sin (x) |

Using Diagonals | A = ½ × d_{1} × d_{2} sin (y) |

**How do you classify a parallelogram? ›**

**If all of the side lengths are equal, then the parallelogram is a square, a rhombus, and a rectangle**. If the side lengths are not all equal, then the parallelogram is a rectangle but is neither a square nor a rhombus.

**How many types of angles are there in a parallelogram? ›**

Answer and Explanation: The two types of angles that a parallelogram always has are **congruent and supplementary angles**.

**What are the properties of a parallelogram in math? ›**

The opposite sides of a parallelogram are equal in measurement and they are parallel to each other. The opposite angles of a parallelogram are equal. The sum of interior angles of a parallelogram is equal to 360°. The consecutive angles of a parallelogram should be supplementary (180°).

**What is the formula for the diagonals of a parallelogram? ›**

The formula for the length of a diagonal of a parallelogram is equal to the magnitude of the resultant of any two adjacent sides. **x = √(a ^{2} + b^{2} – 2ab cos A)** = √(a

^{2}+ b

^{2}+ 2ab cos B)

**Why does the parallelogram formula work? ›**

Discover the magic of geometry! The area of a rectangle and a parallelogram can be calculated in the same way. Just multiply the base by the height! This simple trick works because **a parallelogram can be rearranged into a rectangle**.

**What is the formula for direction in parallelogram? ›**

Parallelogram Law of Vectors Formula

If the resultant vector R makes an angle β with the vector P, then the formulas for its magnitude and direction are: **|R| = √(P ^{2} + Q^{2} + 2PQ cos θ)** β = tan

^{-}

^{1}[(Q sin θ)/(P + Q cos θ)]

### What is the formula for finding the base of a parallelogram? ›

⇒ **Base=AreaHeight**. Q. The base of a parallelogram is twice its height. If the area of the parallelogram is 512 cm^{2}, find the base and height.

**What are the rules of a parallelogram? ›**

**Here are the four properties of a Parallelogram:**

- Opposite angles are equal.
- Opposite sides are equal and parallel.
- Diagonals bisect each other.
- Sum of any two adjacent angles is 180°

**Why are parallelograms important? ›**

The unique properties of the parallelogram have been applied extensively in industry **to accurately transfer mechanical motion from one place to another**. Perhaps the best known application is the pantograph, a four-sided device used by engravers to reproduce an image outline to a user-definable scale.

**What are the elements of a parallelogram? ›**

Elements of parallelogram: Parallelogram is made up of four sides and four angles. There are some terms we usually use to denote parallelogram.

**What is a 4 sided parallelogram called? ›**

Definition: **A quadrilateral** is a polygon with 4 sides. A diagonal of a quadrilat- eral is a line segment whose end-points are opposite vertices of the quadrilateral. In picture below, ABCD is a quadrilateral, AC, BD are the two diagonals.

**What are the 3 special parallelograms? ›**

Special Parallelograms: **Rhombus, Square & Rectangle**.

**What is theorem 4 of parallelogram? ›**

Parallelogram Theorem #4: **The diagonals of a parallelogram bisect each other**.

**Do all parallelograms have 4 parallel sides? ›**

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. Also, the interior angles on the same side of the transversal are supplementary.