Calculating the Area of a Trapezium with Given Side Lengths

In this article, we will explore the method to calculate the area of a trapezium given its parallel and non-parallel sides. We will walk through the steps with a detailed example and provide a general formula for future reference.

Introduction to Trapezium

A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides. These parallel sides are often referred to as the bases of the trapezium, while the non-parallel sides are called the legs.

Example Problem

Consider a trapezium ABCD with the following dimensions:

To find the area of this trapezium, we need to calculate its height.

Step-by-Step Solution

Step 1: Calculate the Height

From the problem, we can draw perpendiculars from one of the vertices of the shorter parallel side to the longer parallel side. This divides the trapezium into a parallelogram and a right triangle. We will use this information to find the height.

Step 2: Use Geometric Properties

Let's drop perpendiculars from points A and B to CD, meeting at points P and Q, respectively. These perpendiculars form right triangles APC and BDQ. The sum of the bases of these triangles equals the difference between the longer and shorter parallel sides:

AC BQ 30 - 20 10

Using the Pythagorean theorem in the right triangles:

AC √(82 - h2) and BQ √(62 - h2)

Therefore:

√(82 - h2) √(62 - h2) 10

Solving for h, we get:

h 24/5 cm

Step 3: Calculate the Area

The area of a trapezium is given by:

Area 1/2 × (sum of bases) × height

Substituting the values:

Area 1/2 × 30 20 × 24/5 120 cm2

General Formula for Trapezium Area

For a trapezium with parallel sides a and b, and non-parallel sides c and d, the area can be calculated using:

Area 1/2 × (a b) × h

Where h is the height, which can be determined using geometric properties. In our example problem, the height was directly determined through the use of right triangles and the Pythagorean theorem.

Conclusion

In the case of the trapezium described, the area is approximately 120 cm2. This method can be applied to any trapezium given its side lengths, allowing for accurate calculation of its area.

If you need to compute the area of a trapezium with specific dimensions, you can follow the same steps. Remember to always verify your calculations with geometric properties and formulas.