Area of a Parallelogram

In mathematics, the concept of area refers to the measurement of space within a two-dimensional shape. One of the important shapes we often encounter in geometry is the parallelogram. In this lesson, we will learn about the area of a parallelogram. Let us learn everything in detail, thereby gaining a comprehensive understanding of this topic. Let us begin our journey by understanding what a parallelogram is, and after that, we will delve into the intricacies of calculating its area.

What is a parallelogram?

A parallelogram is a special type of quadrilateral (a four-sided polygon) whose opposite sides are parallel. This feature of parallel sides defines a parallelogram. Additionally, the opposite sides of a parallelogram are equal in length, and opposite angles are also equal.

In the above illustration, ABCD is a parallelogram with sides AB and CD parallel, and AD and BC parallel.

Basic properties of a parallelogram

• Opposite sides are equal: AB = CD and AD = BC
• Opposite angles are equal: ∠A = ∠C and ∠B = ∠D
• Consecutive angles are supplementary, that is, their sum is 180 degrees.
• The diagonals of a parallelogram bisect each other.

Finding the area of a parallelogram

The area of a parallelogram can be calculated in the same way as a rectangle. Remember, area is the measure of the space inside the shape. For a parallelogram, the area is calculated by multiplying the base by the height. This gives us this basic formula:

Area = base × height

Let us understand this formula in more detail:

Elements of the formula

• Base (B): It refers to the length of one of the sides, usually taken as the base.
• Height (h): This is the perpendicular distance from the base to the opposite side.

Example calculation

Suppose we have a parallelogram with a base of 8 units and a height of 5 units. The area is calculated as follows:

Area = base × height = 8 × 5 = 40 square units

This means that the space inside the parallelogram is 40 square units.

Visualization of parallelogram

To better understand how the area of a parallelogram relates to its shape, imagine rearranging it into a rectangle. One can "cut off" a triangular portion from one side and "slide" it to the opposite side, thereby filling up the space without changing the overall area. This visual helps us understand how multiplying the base by the height effectively measures area.

Applications and practical examples

The concept of the area of a parallelogram is important in real-life applications, such as architecture, engineering, and land measurement. Understanding it can be important in calculating the materials needed for construction, creating design plans, and more.

Consider a flooring project where the floor space is parallelogram-shaped. Calculating the total area allows you to determine the required amount of flooring material. If the base is 10 m and the height is 6 m:

Area = base × height = 10 × 6 = 60 square meters

This shows that 60 square meters of flooring material is required.

Comparing parallelograms to rectangles

Although parallelograms and rectangles appear different at first glance, they share a close geometric relationship. A rectangle is actually a special case of a parallelogram where all the interior angles are right angles (90°). The area of both the formula for determining is essentially the same: base times height. However, in rectangles, the height corresponds directly to one of its sides due to the right angle.

Example comparison

Consider a rectangle and a parallelogram, where both have a base of 6 units and the corresponding height is 4 units. The area is calculated similarly as follows:

Rectangle area = base × height = 6 × 4 = 24 square units
Area of parallelogram = base × height = 6 × 4 = 24 square units

Even though their shapes are different, their areas remain the same within the given dimensions.

Problems and exercises

To solidify your understanding of the area of a parallelogram, practicing a number of problems can be very helpful.

Problem 1:

The base of a parallelogram is 15 m and the height is 8 m. What is its area?

Solution:
Area = base × height = 15 × 8 = 120 square meters

Problem 2:

If the height of a parallelogram is 9 units and the area is 81 square units, what is the base?

Solution:
Area = base × height
81 = base × 9
Base = 81 / 9 = 9 units

Problem 3:

The base of a parallelogram is three times its height. If the area of the parallelogram is 27 square units, find its base and height.

Solution:
Let the height be h units.
Then, the base is 3h units.

Area = base × height
27 = 3h × h
27 = 3h2
h2 = 9
h = 3 units

Base = 3h = 3 × 3 = 9 units

Conclusion

Understanding the area of a parallelogram is essential for solving various mathematical problems and handling real-world tasks involving location and measurement. Whether you are checking architectural plans or calculating land for development, this is a great way to know how to calculate the area of a parallelogram. Mastering the concept gives you an essential mathematical tool. Remember, the simple multiplication of the base and height leads you to the area, leading to an efficient way to measure space inside parallelogram shapes.

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