Calculating the Area of a Trapezium with Given Side Lengths

Trapezia, also known as trapezoids, are quadrilaterals with one pair of parallel sides. Calculating the area of a trapezium can be straightforward if you know the lengths of its parallel and non-parallel sides. This article will walk you through a detailed process to determine the area of a trapezium with given side lengths, utilizing both Heron's and Brahmagupta's formulas.

Rewriting the Given Problem

We are given the following side lengths for a trapezium: One parallel side: a 60 cm The other parallel side: b 77 cm One non-parallel side: c 25 cm The other non-parallel side: d 26 cm

Step-by-Step Calculation

The area of a trapezium can be calculated using the formula:

Area 1/2 * (a b) * h

Where h is the height of the trapezium. In our case, since we do not have the height directly, we will use an alternative method to find the height.

Step 1: Calculate the Semi-Perimeter

The semi-perimeter s is given by:

s (a b c d) / 2

Substituting the given values:

s (60 77 25 26) / 2 188 / 2 94 cm

Step 2: Use Brahmagupta's Formula

Brahmagupta's formula for the area of a cyclic quadrilateral (if the trapezium can be inscribed in a circle) can also be used to find the height. First, we need to find the expressions for s - a, s - b, s - c, and s - d.

s - a 94 - 60 34

s - b 94 - 77 17

s - c 94 - 25 69

s - d 94 - 26 68

The formula for the area is:

Area √[(s - a) * (s - b) * (s - c) * (s - d)]

Substituting the values:

Area √[34 * 17 * 69 * 68]

Step 3: Calculate the Area

First, calculate the products:

34 * 17 578

69 * 68 4692

Now, multiply these results:

578 * 4692 2712456

Finally, take the square root:

Area ≈ √2712456 ≈ 1646.3 cm2

Alternative Method

An alternative method to find the area involves dropping perpendiculars from the vertices of the non-parallel sides to the longer parallel side. This forms a rectangle and two right-angled triangles. Utilizing trigonometric functions or the Pythagorean theorem can help determine the height.

Let's use the given alternative method:

h height of the trapezium.

From the given equation:

√(262 - h2)√(252 - h2) 77 - 60 17

Multiply both sides by the conjugate:

676 - 625 17√(262 - h2) - √(252 - h2)

17 √(262 - h2) - √(252 - h2)

Add this to the equation:

1 2 √(262 - h2) 10

Solve for h:

676 - h2 100

h2 676 - 100 576

h 24 cm

Finally, calculate the area:

Area (1/2) * (60 77) * 24 1644 cm2

Conclusion

The area of a trapezium with the given side lengths can be calculated using various methods, including Brahmagupta's and Heron's formulas. In this case, the area is approximately 1646.3 cm2 or 1644 cm2 depending on the method used. This detailed step-by-step guide will help you understand and calculate the area of a trapezium accurately.