Understanding the concept of area in mathematics is essential for various real-world applications. In mathematics this article, we will delve into the topic of finding the area of parallelograms. Parallelograms are fascinating geometric shapes with unique properties, and being able to calculate their area accurately is a valuable skill. We will explore the formulas, step-by-step calculation methods, practical applications, common mistakes, and key insights related to the area of parallelograms.

What is a Parallelogram?

A parallelogram is a special type of shape with four sides that are formed by lines that never cross. The angles inside a parallelogram can be different, but what makes it a parallelogram is that the opposite sides are both parallel and the same length. So, if a four-sided shape has opposite sides that are both parallel and the same length, it is called a parallelogram.

Observe the following figure which shows the three types of parallelograms:

What is the Area of a Parallelogram?

The area of a parallelogram is the measure of the two dimensional region enclosed by the sides of a parallelogram. It tells us how many little squares can fit inside it. We measure it using square units like square centimeters or square meters. The area is the space inside the parallelogram, which is a shape with four sides that are parallel in pairs.

In our everyday lives, we encounter various shapes that aren't just rectangles or squares. While rectangles and parallelograms have some similarities in their properties, the way we calculate the area of a rectangle is similar to how we calculate the area of a parallelogram.

Area of Parallelogram Formula

To find the area of a parallelogram, you multiply the length of its base by the height. The base and height of a parallelogram are at right angles to each other, as shown in the accompanying diagram. Therefore, the formula to calculate the area of a parallelogram is as follows:

Area of parallelogram \(= b \times h\) square units

Where, 

• \(b =\) length of the base
• \(h =\) height or altitude

Let's understand the formula by looking at an example. Imagine we have a parallelogram called \(PQRS\). We can use grid paper to find its area by counting the squares.

Next, we will find the area of a parallelogram using two different methods: 

(i). by counting squares, and 

(ii). by using a formula.

Area of Parallelogram by Counting Squares:

Based on the figure shown:

• The number of full squares is \(16\).
• The number of half squares is \(8\).
• To find the area of the parallelogram, we add the number of full squares and half of the number of half squares. This gives us \(16 + \left(\frac{1}{2}\right) \times 8 = 16 + 4 = 20\) square units.

Area of Parallelogram by Using Formula:

Additionally, if we look at the figure, we notice that \(ST\) is perpendicular to \(PQ\). By counting the squares, we find that:

• The length of side \(PQ\) is \(5\) units.
• The corresponding height, \(ST\), is \(4\) units.
• To find the area of the parallelogram, we multiply the side length \((5)\) by the height \((4)\), resulting in \(20\) square units.

It's important to note that we obtained the same answer using both methods. Hence, we can conclude that the area of the given parallelogram is equal to the base multiplied by the altitude.

Let's engage in an activity to grasp the concept of parallelogram area.

Step I: Start by drawing a parallelogram \((PQRS)\) on a piece of cardboard and mark a line \((SE)\) that represents the altitude. Cut along this line. 

Step II: Next, cut out the triangular section \((PSE)\) from the parallelogram. 

Step III: Take a white chart and glue the remaining portion of the parallelogram \((EQRS)\) onto it. 

Step IV: Finally, paste the triangular section \((PSE)\) onto the white chart, aligning its sides \(RQ\) and \(SP\).

During our investigation, we noticed that the area of a rectangle is the same as the area of a parallelogram. Moreover, we discovered that the base and height of the parallelogram correspond to the length and width of the rectangle, respectively.

In other words, to calculate the area of a parallelogram, you can simply multiply its base by its height. Interestingly, this formula is equivalent to multiplying the length and width of a rectangle.

How to Calculate the Area of Parallelogram?

The area of a parallelogram can be calculated in the following situations:

• When you know the length of the base and the height.
• When you know the length of the adjacent sides and the angle between them.
• When you know the lengths of both diagonals and one of the angles where they intersect.

Area of Parallelogram Using Height

Suppose we have a parallelogram with two parallel sides '\(a\)' and '\(b\)' and a height '\(h\)' (which is the distance between '\(a\)' and '\(b\)' in a straight line). In this case, the formula to find the area of the parallelogram is:

\(Area = Base \times Height\)

Using the formula, we can calculate the area \((A)\) by multiplying the length of the base \((b)\) by the height \((h)\):

\(A = b \times h\) square units

Example: Let's say we have a parallelogram with a base of \(10\) cm and a height of \(8\) cm. We want to find its area.

Solution: Given that the length of the base is \(10\) cm and the height is \(8\) cm.

Using the formula for the area of a parallelogram, we multiply the base \((10)\) by the height \((8)\) to get the area: 

\(Area = 10 \times 8 = 80\) \(cm^{2}\).

Area of Parallelogram Using Lengths of Sides

The area of a parallelogram can be found even if we don't know the height. Instead, if we have the lengths of the adjacent sides and the angle between them, we can use a different formula.

The formula is:

Area = (length of side \(a\)) \(\times\) (length of side \(b\)) \(\times \sin(\theta)\).

Here, side \(a\) and side \(b\) refer to the lengths of the adjacent sides, and \(\theta\) represents the angle between those sides.

By applying this formula, we can calculate the area of a parallelogram without needing the height.

Example: Suppose we have a parallelogram with two adjacent sides measuring \(4\) units and \(6\) units, and the angle between them is \(90\) degrees. Let's find the area of this parallelogram.

Solution: Given: \(a = 4\) units, \(b = 6\) units, \(\theta = 90\) degrees 

Using the formula for the area of a parallelogram, 

\(Area = ab \sin(\theta)\) 

Plugging in the values, 

\(A = 4 \times 6 \times \sin(90^{\circ})\)

\(A = 24 \times \sin(90^{\circ})\)

\(A = 24 \times 1 = 24\) square units.

Important Note: When the angle between the sides of a parallelogram measures \(90\) degrees, the parallelogram transforms into a rectangle.

Area of Parallelogram Using Diagonals

To find the area of a parallelogram, you can use the lengths of its diagonals. Diagonals are lines that connect opposite corners of the parallelogram. If we know the length of the diagonals and the angle between them, denoted as \(x\), we can use the following formula:

\(Area = \frac{1}{2} \times d_1 \times d_2 \times \sin(x)\)

Here, \(d_1\) and \(d_2\) represent the lengths of the diagonals, and \(x\) is the angle between them.

Area of Parallelogram in Vector Form

You can find the area of a parallelogram using different formulas, even if you have the sides or diagonals in vector form.

• Formula 1: If you have the adjacent sides \(a\) and \(b\) as vectors, the area is given by the magnitude of the cross product of \(\textbf{a}\) and \(\textbf{b}\), denoted as \(\left|\textbf{a}\times\textbf{b}\right|\).
• Formula 2: If you know the diagonals \(\textbf{d1}\) and \(\textbf{d2}\) as vectors, the area can be calculated as half the magnitude of the cross product of \(\textbf{d1}\) and \(\textbf{d2}\), denoted as \(\frac{1}{2}\left|(\textbf{d1}\times\textbf{d2})\right|\).

Both formulas involve the "cross product of vectors" denoted by "\(\times\)". These two formulas are equivalent, and we'll explore their relationship in detail.

Area of Parallelogram Solved Examples

1) Consider a parallelogram with a base of \(6\) units and a height of \(4\) units. Calculate its area.

Solution)

Using the formula for the area of a parallelogram,

\(A = base \times height\), we can substitute the given values:

\(A = 6 \times 4 = 24\) square units

Therefore, the area of the parallelogram is \(24\) square units.

2) Given the vectors \(\textbf{a} = (2, 3)\) and \(\textbf{b} = (4, 1)\), find the area of the parallelogram formed by these vectors.

Solution)

To calculate the area using vector notation, we use the formula \(\left|\textbf{a}\times\textbf{b}\right|\), where \(\times\) denotes the cross product.

First, find the cross product of vectors \(\textbf{a}\) and \(\textbf{b}\):

\(\textbf{a} \times \textbf{b} = (2, 3) \times (4, 1)\)

\(\textbf{a} \times \textbf{b} = (2\times 1) - (3\times 4) = 2 - 12 = -10\)

Next, take the magnitude of the cross product:

\(\left|\textbf{a}\times\textbf{b}\right| = \left|-10\right| = 10\)

Therefore, the area of the parallelogram formed by vectors \(a\) and \(b\) is \(10\) square units.

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.