Area of Trapezium: A Detailed Explanation

Imagine a shape that's part rectangle, part triangle, with a unique twist—this is the trapezium. Unlike regular shapes, a trapezium has two parallel sides of different lengths. This makes its area calculation both interesting and essential for various practical applications.  

However, we cannot always find the Area of a Trapezium by drawing unit squares. Using a standard formula would help in this situation. Read this blog to explore the Area of a Trapezium and the formula to calculate it. 

Table of Contents 

1) What is a Trapezium? 

2) Different Types of Trapezium  

3) Area of Trapezium Formula 

4) Area of a Trapezium with a Parallelogram 

5)  Area of a Trapezium with a Triangle 

6) How to Find the Area of a Trapezium> 

7) Examples of the Area of a trapezium 

8) Conclusion 

What is a Trapezium?
 

A trapezium is a four-sided polygon that falls under the category of quadrilaterals. It is defined as having only one set of parallel sides. The trapezium's area is the space inside its boundaries, measured in square units like cm², m², mm², etc. 

Different Types of Trapezium  

Trapezia can be categorised according to their characteristics, like the lengths of their sides or the measurements of their angles. 

1) An Isosceles Trapezoid has equal base angles, and its diagonals have equal lengths as well. 

2) Scalene Trapezium has different base angles and diagonal lengths. 

Area of Trapezium Formula 

Let's calculate the area of a trapezium with the help of two parallel sides and the height of the trapezium. The formula for calculating the area (A) of a trapezium is:

where: 

1) (a) and (b) are the lengths of the parallel sides (bases), 

2) (h) is the height (the perpendicular distance between the bases). 

There are two methods to derive this formula: 

1) Using a parallelogram: By rearranging two identical trapeziums to form a parallelogram, you can derive the area formula. 

2) Using a triangle: By dividing the trapezium into two triangles and calculating their combined area, you can also derive the formula.  

Area of a Trapezium with a Parallelogram 

Consider two identical trapeziums with bases (a) and (b) and height (h). Let us assume (A) as the area of each trapezium. Now, turn the second trapezium upside down and attach it to the first one. 

By joining the two trapeziums, we can form a parallelogram with a base of (a + b) and a height of (h). 

1) Area of a parallelogram is base × height = (a + b) h. 

2) Area of the above parallelogram in terms of 'A' is, A + A = 2A. 

Thus, 2A = (a + b) h 

⇒ A = (a+b)h/2 

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Area of a Trapezium with a Triangle 

Follow these steps to get the formula for the area of a trapezium: 

Step 1: Divide one of the non-parallel sides of the trapezium into two equal parts. 

Step 2: Cut out a triangular section from the trapezium. 

Step 3: Attach this triangular portion to the bottom of the shape. 

Now, the trapezium is rearranged to form a triangle, and the areas of both shapes are identical. The base of the triangle is (a + b), while the height remains h.  

Thus, the area of the trapezium equals the area of the triangle, calculated as: 

Area of Trapezium = Area of triangle = ½ × base × height = ½ (a + b) × h. 

How to Find the Area of a Trapezium 

We can derive the area formula for a trapezium by relating it to a triangle. Consider a trapezium with parallel sides of lengths a and b, and a height of h. To derive the formula: 

Step 1: Divide one of the trapezium's legs into two equal parts. 

Step 2: Cut a triangular section from the trapezium. 

Step 3: Attach this triangle to the bottom of the trapezium (as shown in the diagram). 

This rearrangement transforms the trapezium into a triangle. From the diagram, it's clear that the areas of the original trapezium and the newly formed triangle are the same. The triangle’s base is now equal to (a + b), and the height remains h. 

Thus, the area of the trapezium is the same as the area of the triangle, which can be calculated using the formula: 

Area of the trapezium = Area of the triangle = ½ × base × height = ½ × (a + b) × h. 

Examples of the Area of a Trapezium 

Below are several examples illustrating different scenarios for calculating the area of a trapezium. 

Example 1: Determining the Area with Known Parallel Side Lengths and Height

If the lengths of the parallel sides are a = 8 units and b = 5 units, and the height h = 4 units, the area of the trapezium can be calculated using the formula: 

Example 2: Finding the Area of a Trapezium with Unit Conversions

In this example, let’s say the parallel sides of the trapezium are a = 500 cm and b = 300 cm, and the height is 2 meters. First, convert the height to centimetres: 

Example 3: Calculating the Height When the Area is Given

Suppose the area of the trapezium is 45 square units, and the lengths of the parallel sides are a = 5 units and b = 3 units. To find the height h, use the formula and solve for h: 

Example 4: Finding the Base Length When the Area is Known

If the area of the trapezium is 64 square units, and the height is h = 4 units, with one of the parallel sides a = 10 units, find the other side b. Use the area formula and solve for b: 

These examples explain how to calculate the area of a trapezium in various scenarios without values of side lengths, height, and area. 

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Conclusion 

The area of a trapezium can be easily calculated using the formula ½ × (a + b) × h. Here, (a) and (b) are the lengths of the parallel sides, and h is the height. Whether you know the sides, height, or area, this formula provides a simple way to solve various geometric problems involving trapeziums. 

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