Formula for the Area of a Trapezium

Oct 13, 2025 by wbrum.org 142 views

Introduction

In geometry, a trapezium is a quadrilateral with at least one pair of parallel sides. The area of a trapezium can be calculated using the formula: Area = (1/2) × (sum of parallel sides) × (perpendicular distance between them). In this article, we will use this formula to find the length of each parallel side of a trapezium given its area and the ratio of its parallel sides.

Understanding the Problem

The problem states that the area of the trapezium is 384 cm², and the ratio of its parallel sides is 3:5. Let's assume that the lengths of the parallel sides are 3x and 5x, respectively. The perpendicular distance between the parallel sides is given as 12 cm.

Formula for the Area of a Trapezium

The formula for the area of a trapezium is:

Area = (1/2) × (sum of parallel sides) × (perpendicular distance between them)

In this case, the sum of the parallel sides is 3x + 5x = 8x. Substituting the given values, we get:

384 = (1/2) × 8x × 12

Solving for x

To find the value of x, we can simplify the equation:

384 = 48x

Dividing both sides by 48, we get:

x = 384/48 x = 8

Finding the Length of Each Parallel Side

Now that we have the value of x, we can find the length of each parallel side:

Length of the shorter side = 3x = 3 × 8 = 24 cm Length of the longer side = 5x = 5 × 8 = 40 cm

Conclusion

In this article, we used the formula for the area of a trapezium to find the length of each parallel side given its area and the ratio of its parallel sides. We assumed that the lengths of the parallel sides are 3x and 5x, respectively, and used the given values to solve for x. Finally, we found the length of each parallel side by substituting the value of x into the expressions for the lengths of the parallel sides.

Example Problems

Problem 1

The area of a trapezium is 420 cm², and the ratio of its parallel sides is 2:3. Find the length of each parallel side.

Solution

Let's assume that the lengths of the parallel sides are 2x and 3x, respectively. The perpendicular distance between the parallel sides is given as 15 cm.

Using the formula for the area of a trapezium, we get:

420 = (1/2) × (2x + 3x) × 15

Simplifying the equation, we get:

420 = 25x

Dividing both sides by 25, we get:

x = 420/25 x = 16.8

Now that we have the value of x, we can find the length of each parallel side:

Length of the shorter side = 2x = 2 × 16.8 = 33.6 cm Length of the longer side = 3x = 3 × 16.8 = 50.4 cm

Problem 2

The area of a trapezium is 360 cm², and the ratio of its parallel sides is 4:6. Find the length of each parallel side.

Solution

Let's assume that the lengths of the parallel sides are 4x and 6x, respectively. The perpendicular distance between the parallel sides is given as 10 cm.

Using the formula for the area of a trapezium, we get:

360 = (1/2) × (4x + 6x) × 10

Simplifying the equation, we get:

360 = 50x

Dividing both sides by 50, we get:

x = 360/50 x = 7.2

Now that we have the value of x, we can find the length of each parallel side:

Length of the shorter side = 4x = 4 × 7.2 = 28.8 cm Length of the longer side = 6x = 6 × 7.2 = 43.2 cm

Final Thoughts

Introduction

In the previous article, we discussed how to find the length of each parallel side of a trapezium given its area and the ratio of its parallel sides. In this article, we will answer some frequently asked questions (FAQs) about the area of a trapezium.

Q: What is the formula for the area of a trapezium?

A: The formula for the area of a trapezium is:

Area = (1/2) × (sum of parallel sides) × (perpendicular distance between them)

Q: How do I find the length of each parallel side of a trapezium?

A: To find the length of each parallel side of a trapezium, you need to know the area of the trapezium and the ratio of its parallel sides. Let's assume that the lengths of the parallel sides are 3x and 5x, respectively. The perpendicular distance between the parallel sides is given as 12 cm. Using the formula for the area of a trapezium, you can solve for x and then find the length of each parallel side.

Q: What if the ratio of the parallel sides is not given?

A: If the ratio of the parallel sides is not given, you can use the formula for the area of a trapezium to find the length of each parallel side. However, you will need to know the lengths of the parallel sides or the ratio of their lengths.

Q: Can I use the formula for the area of a trapezium to find the length of each parallel side if the trapezium is not a right trapezium?

A: No, the formula for the area of a trapezium is only applicable to right trapeziums. If the trapezium is not a right trapezium, you will need to use a different formula to find the length of each parallel side.

Q: How do I find the area of a trapezium if I know the length of each parallel side?

A: To find the area of a trapezium if you know the length of each parallel side, you can use the formula:

Area = (1/2) × (sum of parallel sides) × (perpendicular distance between them)

Q: What if the perpendicular distance between the parallel sides is not given?

A: If the perpendicular distance between the parallel sides is not given, you will need to know the lengths of the parallel sides or the ratio of their lengths to find the area of the trapezium.

Q: Can I use the formula for the area of a trapezium to find the length of each parallel side if the trapezium is a rectangle?

A: No, the formula for the area of a trapezium is only applicable to trapeziums. If the trapezium is a rectangle, you will need to use a different formula to find the length of each parallel side.

Q: How do I find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a fraction?

A: To find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a fraction, you can use the formula:

Area = (1/2) × (sum of parallel sides) × (perpendicular distance between them)

Q: What if the ratio of the parallel sides is given as a decimal?

A: If the ratio of the parallel sides is given as a decimal, you can convert it to a fraction and then use the formula for the area of a trapezium to find the length of each parallel side.

Q: Can I use the formula for the area of a trapezium to find the length of each parallel side if the trapezium is an isosceles trapezium?

A: Yes, the formula for the area of a trapezium is applicable to isosceles trapeziums. However, you will need to know the lengths of the parallel sides or the ratio of their lengths to find the area of the trapezium.

Q: How do I find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a ratio of integers?

A: To find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a ratio of integers, you can use the formula:

Area = (1/2) × (sum of parallel sides) × (perpendicular distance between them)

Q: What if the ratio of the parallel sides is given as a ratio of decimals?

A: If the ratio of the parallel sides is given as a ratio of decimals, you can convert it to a ratio of integers and then use the formula for the area of a trapezium to find the length of each parallel side.

Q: Can I use the formula for the area of a trapezium to find the length of each parallel side if the trapezium is a right trapezium with one pair of parallel sides of equal length?

A: Yes, the formula for the area of a trapezium is applicable to right trapeziums with one pair of parallel sides of equal length. However, you will need to know the lengths of the parallel sides or the ratio of their lengths to find the area of the trapezium.

Q: How do I find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a ratio of integers with a common factor?

A: To find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a ratio of integers with a common factor, you can use the formula:

Area = (1/2) × (sum of parallel sides) × (perpendicular distance between them)

Q: What if the ratio of the parallel sides is given as a ratio of decimals with a common factor?

A: If the ratio of the parallel sides is given as a ratio of decimals with a common factor, you can convert it to a ratio of integers and then use the formula for the area of a trapezium to find the length of each parallel side.

Q: Can I use the formula for the area of a trapezium to find the length of each parallel side if the trapezium is a right trapezium with one pair of parallel sides of equal length and the ratio of the parallel sides is given as a ratio of integers?

A: Yes, the formula for the area of a trapezium is applicable to right trapeziums with one pair of parallel sides of equal length and the ratio of the parallel sides is given as a ratio of integers. However, you will need to know the lengths of the parallel sides or the ratio of their lengths to find the area of the trapezium.

Q: How do I find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a ratio of integers with a common factor and the trapezium is a right trapezium with one pair of parallel sides of equal length?

A: To find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a ratio of integers with a common factor and the trapezium is a right trapezium with one pair of parallel sides of equal length, you can use the formula:

Area = (1/2) × (sum of parallel sides) × (perpendicular distance between them)

Q: What if the ratio of the parallel sides is given as a ratio of decimals with a common factor and the trapezium is a right trapezium with one pair of parallel sides of equal length?

A: If the ratio of the parallel sides is given as a ratio of decimals with a common factor and the trapezium is a right trapezium with one pair of parallel sides of equal length, you can convert it to a ratio of integers and then use the formula for the area of a trapezium to find the length of each parallel side.

Q: Can I use the formula for the area of a trapezium to find the length of each parallel side if the trapezium is a right trapezium with one pair of parallel sides of equal length and the ratio of the parallel sides is given as a ratio of integers with a common factor?

A: Yes, the formula for the area of a trapezium is applicable to right trapeziums with one pair of parallel sides of equal length and the ratio of the parallel sides is given as a ratio of integers with a common factor. However, you will need to know the lengths of the parallel sides or the ratio of their lengths to find the area of the trapezium.

Q: How do I find the length of each parallel side of a trapezium if the area is given in square units and the ratio of the parallel sides is given as a ratio of integers with a common factor and the trapezium is a right trapezium with one pair of parallel sides of equal length?

A: To find the length of each parallel side of a trapezium if the area is given