A parallelogram is a four-sided shape whose opposite sides are equal and parallel. Some common parallelograms are rectangles, squares, and rhombuses. Knowing about parallelograms—its properties, area, perimeter, and formulas for diagonals—is crucial to solve geometry problems in US competitive exams such as the SAT. Learning these makes it easy for students to recognize shapes at sight, use formulas, and crack problems easily, hence scoring better in the Mathematics section. This article delves into the nature, characteristics, and equations of parallelograms with elaborative examples to enhance your problem-solving abilities.

Parallelogram

A two-dimensional geometrical shape in mathematics whose opposite sides are parallel to one another is said to be a parallelogram. The word ‘parallelogram’ was taken from the Greek word ‘parallelogrammon’ which means “bounded by parallel lines”. In such a shape the angles between the adjacent sides can differ but;

• The measure of the opposite angles is equal.
• Also, the opposite sides are always equal and parallel.

A representation of the same is shown below:

Parallelogram Shaped Objects

Some of the real-life parallelogram shape examples are as follows:

• The solar panel is commonly used as a renewable source and is usually fabricated in a rectangular or parallelogram shape. These shapes in the solar panel are most preferred as the installation of such shapes on the roof terrace is preferably easy.
• The common structure of the staircase is also parallelogram-shaped.
• The desk that we either use for study or to work, the dinner table, dressing table and other such related furniture that we use in our day-to-day work are rectangular in shape.
• Tiles used for covering the floor, ground, etc come in different shapes and sizes. Some of the popular shapes are rhombus, parallelogram, square, and rectangle.
• The saltbox roof, shed roof or gable roof of a cottage when observed from the side view resembles the parallelogram shape.
• In addition to the above examples, nowadays even architects use different parallelogram patterns to create unique building designs. Also, several stationery things like erasers, sharpeners, rulers, etc are parallelogram-shaped.

Types of Parallelogram

When it comes to geometric shapes, parallelograms are categorized into squares, rectangles, and rhombus. Each of these shapes has its respective properties. In the previous two headers, we read about the parallelogram definition and different real-life examples of it in geometry. Moving forward let us understand the various types:

• Rectangle
• Square
• Rhombus

Rectangle

A rectangle is a type of parallelogram wherein the measure of one angle is equal to 90 degrees. Some of the common properties of rectangles are;

• There are two diagonals that bisect each other.
• The opposite side is equal and parallel to one another.
• The adjacent sides meet each other at 90 degrees.

Square

A square is a rectangle in which all its sides are of equal length. Some of the common properties of squares are.

• All the 4 interior angles are 90 degrees each.
• Also, the diagonals bisect one another at 90 degrees.

Rhombus

A rhombus is another type of parallelogram where the four sides are equal. Also, the opposite angles are equal. Some properties related to rhombus are as follows:

• The opposite angles are of equal length.
• The diagonals divide one another perpendicularly.

As of now, we studied the three different types of parallelograms. There is one more special case namely rhomboid.

Rhomboid

A rhomboid is a special case of parallelogram wherein the adjacent sides are of unequal lengths and the angles are not right-angled. Also, it is the most recognised parallelogram shape. Some of the important features of the rhomboid shape are:

• The opposite pair of sides are equal and parallel.
• The opposite angles in a rhomboid are also equal.
• The diagonal splits the rhomboid into 2 equal triangles.
• The sum of all the interior angles is equivalent to 360 degrees.

Parallelogram Formula

So far we understood that a polygon where the pair of parallel sides are identical in length is called a parallelogram. Let us understand the formula for area, perimeter, diagonals and angles.

Area of Parallelogram

The area of a parallelogram is the space confined between its four sides. The area of a parallelogram can be calculated using the length of its base and height.

If we are given a parallelogram ABCD, such that AB is the base, and h is the height, then;

Area of the parallelogram = Base × Height = (b)×(h)

The area is measured in square units. The commonly used units are \(\text{cm}^2\), \(\text{m}^2\), \(\text{inch}^2\) and so on.

Learn more about Mensuration 3D here.

Perimeter of Parallelogram

The perimeter of any closed-shaped object is the total length of the boundary. Therefore the perimeter of a parallelogram is the measurement of the length of its outline that is the sum of all its sides. As the opposite sides are equal so the formula for it becomes:

Perimeter of parallelogram formula = P = 2(length + breadth)= 2 (a + b) units.

The perimeter is calculated in units like cm, m, inch and so on.

Diagonals of a Parallelogram

The diagonals of a parallelogram can be drawn by joining the two non-adjacent vertices of the parallelogram. A parallelogram has two diagonals as shown in the figure.

The diagonals of a parallelogram:

\(x=\sqrt{p^2+q^2-2pq\cos A}\text{ or }x=\sqrt{p^2+q^2+2pq\cos B}\)

\(y=\sqrt{p^2+q^2+2pq\cos A}\text{ or }y=\sqrt{p^2+q^2-2pq\cos B}\)

The sum of the square of the diagonals=\(x^2+y^2=2\left(p^2+q^2\right)\).

Angles in a Parallelogram

As of now, we studied the definition and formula for area, perimeter and diagonals. One more thing that is important in any type of polygon is the angle made by the adjacent sides.

In a parallelogram, ABCD, the ∠A, ∠B, ∠C and ∠D are respective 4 angles.

The opposite angles of a parallelogram are identical in measure. That is for the above figure we can say that ∠D = ∠B and ∠A = ∠C.

Also, the sum of angles of a parallelogram is equal to 360°. That is according for the above diagram ∠A + ∠B + ∠C + ∠D = 360°.

Properties of Parallelogram

Now that you know what a parallelogram is, its types, formulas and definitions, let us understand the various properties related to the topic.

• The opposite sides of a parallelogram are equal and parallel.

• The adjacent angles of a parallelogram are unequal. Or can also say the opposite angles of a parallelogram are equal in measure.

• The sum of the adjacent angles in a parallelogram is equal to 180 degrees.
• There are two diagonals in a parallelogram and each of them bisects one another.
• The diagonals divide the parallelogram into two congruent triangles.

• One diagonal divides the parallelogram into two equal triangles. Also, two diagonals divide the parallelogram into 4 equal triangles.

Difference between Rectangle and Parallelogram

When it comes to the different types of a parallelogram, it all rolls down to the parameter and properties to identify their shape and name. There are two shapes rectangle and parallelogram that are very similar when viewed. Let us understand the difference between them with the given tabular data:

Solved Examples on Parallelogram

Some solved examples related to the topic are covered under this header. Although the solution is given, try to solve them first and verify your answer with the solution.

Solved Example 1: Determine the area and perimeter of a parallelogram whose two sides i.e. base length=7cm and width=5 cm. Also, the height is 10 cm.

Solution: Given, Base length = 7 cm and Height = 10 cm.

Using the formula for the area and perimeter of a parallelogram:

Area = Base x Height

Area = 7 × 10

Area = 70 Sq.cm

P =2 (length + breadth)

P =2 (7cm + 5 cm)

P =24

Solved Example 2: If in a given parallelogram PQRS, ∠R = 90°. Obtain the values for the other angle.

Solution: The diagram for the PQRS is as shown:

According to the properties, the opposite angles are equal.

Thus, ∠R = ∠P = 90°

Using the supplementary angle property:

The adjacent angles of a parallelogram are equal to 180°.

∠Q+∠R = 180°

⇒ ∠Q = 180°− 90° (As ∠R = 90°)

Hence, ∠Q = 90°

As the sum of all the angles in a parallelogram is equal to 360°.

∠P + ∠Q + ∠R + ∠S=360°

Substituting the values we get ∠S=90°.

That is the measure of all the angles for the given figure is equal to 90 degrees.

Note: When one of the angles of a parallelogram is equal to 90 degrees, then all the other angles are also equal to 90 degrees..

Solved Example 3: Which of the following shapes are not counted under parallelogram?

• Square
• Rectangle
• Trapezium
• Rhombus

Solution: From the given options, the trapezium is not counted under a parallelogram. The condition for a shape to be a parallelogram is that its opposite sides are always equal and parallel.

But in a trapezium there is only one pair of parallel sides, also both the pair of opposite sides are not equal.

Solved Example 4: In a parallelogram LMNO, if ∠L = 80°, Obtain the value of ∠O.

Solution: The image below shows the LMNO parallelogram, where ∠L = 80° and we need to determine the value of ∠O.

The opposite angles in a parallelogram shape are equal. Hence ∠L = ∠N = 80° also, ∠M = ∠O.

Also, ∠L + ∠M + ∠N + ∠O= 360°

80° + ∠M + 80° + ∠O = 360°

∠M + ∠O = 200°

As ∠M = ∠O (using the parallelogram property)

2 ∠O = 200°

Therefore, ∠O = 100°.

We hope this article enhances your understanding of parallelograms and boosts your exam preparation. Mastering these concepts is crucial for excelling in US competitive exams like the SAT, where geometry questions often test your knowledge of shapes, properties, and formulas. For more updates on math topics and other subjects, stay connected with the Testbook App. You can also explore test series to assess your knowledge and sharpen your problem-solving skills for various exams.