1. Area of a Trapezium
A trapezium (called a trapezoid in some countries) is a quadrilateral with one pair of opposite sides parallel. These parallel sides are called the bases of the trapezium.
The area of a trapezium tells us the total space enclosed between its sides. To calculate this, we mainly use the lengths of its parallel sides and the height (the perpendicular distance between them).
1.1. Parallel Sides and Height
Let the lengths of the two parallel sides be \(a\) and \(b\). The height (\(h\)) is the perpendicular distance between these sides, not the slant distance.
This height is essential because without a right-angle distance, the area cannot be measured correctly.
2. Area Formula
The area of a trapezium is given by the formula:
\( A = \dfrac{1}{2}(a + b)h \)
where:
- \(a\) and \(b\) are the lengths of the parallel sides
- \(h\) is the perpendicular height
2.1. Why the Formula Works
The area of a trapezium is the average of the parallel sides multiplied by the height. This works because the shape can be broken or rearranged into a rectangle-like shape whose width equals the average of \(a\) and \(b\).
Another way to understand this is by splitting the trapezium into a rectangle plus two right triangles.
3. Examples
Let’s see how the formula is applied in simple cases.
3.1. Example 1
A trapezium has parallel sides \(a = 10\,\text{cm}\) and \(b = 6\,\text{cm}\), and height \(h = 5\,\text{cm}\). Its area is:
\( A = \dfrac{1}{2}(a + b)h = \dfrac{1}{2}(10 + 6) \times 5 = 8 \times 5 = 40\,\text{cm}^2 \)
3.2. Example 2
A trapezium-shaped garden has parallel sides of \(20\,\text{m}\) and \(12\,\text{m}\), with height \(h = 15\,\text{m}\). Its area is:
\( A = \dfrac{1}{2}(20 + 12) \times 15 = \dfrac{1}{2} \times 32 \times 15 = 16 \times 15 = 240\,\text{m}^2 \)