Find the area of trapezium with parallel sides 8cm and 12cm and height 5cm

Question

Find the area of a trapezium with parallel sides of length 8 cm and 12 cm, and a height (perpendicular distance between the parallel sides) of 5 cm.

Solution — Step by Step

A trapezium (called a trapezoid in American English) is a quadrilateral with exactly one pair of parallel sides.

The formula for the area of a trapezium is:

\text{Area} = \frac{1}{2} \times (a + b) \times h

where $a$ and $b$ are the lengths of the two parallel sides (called the “bases”), and $h$ is the perpendicular height between them.

From the problem:

First parallel side: $a = 8$ cm
Second parallel side: $b = 12$ cm
Height: $h = 5$ cm

\text{Area} = \frac{1}{2} \times (8 + 12) \times 5

= \frac{1}{2} \times 20 \times 5

= \frac{1}{2} \times 100

= \boxed{50 \text{ cm}^2}

Why This Works

A trapezium can be cut into a rectangle and two triangles. The average of the two parallel sides gives the width of an equivalent rectangle, and multiplying by the height gives the area:

\text{Area} = \underbrace{\frac{a + b}{2}}_{\text{average base}} \times h

Here, the average base = $(8+12)/2 = 10$ cm. Multiplying by height $5$ cm gives $50$ cm².

This formula also works for parallelograms (where $a = b$ , giving $\text{Area} = a \times h$ ) and triangles (where $b = 0$ , giving $\text{Area} = \frac{1}{2} ah$ ). The trapezium formula is the general form.

Alternative Method

You can split the trapezium into simpler shapes:

Option 1: Draw a diagonal to split into two triangles:

Triangle 1: base = 8 cm, height = 5 cm → Area = $\frac{1}{2} \times 8 \times 5 = 20$ cm²
Triangle 2: base = 12 cm, height = 5 cm → Area = $\frac{1}{2} \times 12 \times 5 = 30$ cm²
Total = 50 cm² ✓

Option 2: Draw a perpendicular from one end of the shorter side to the longer side, splitting into a rectangle (8 × 5) and a triangle. This works only for a right trapezium.

Common Mistake

Students sometimes use the formula $\text{Area} = \text{base} \times \text{height}$ (for a rectangle or parallelogram) and multiply one parallel side by the height: $8 \times 5 = 40$ or $12 \times 5 = 60$ . Both are wrong. For a trapezium, you must use the average of the two parallel sides: $(8+12)/2 = 10$ . The formula $\frac{1}{2}(a+b)h$ accounts for the fact that the trapezium is neither as wide as its longer side nor as narrow as its shorter side throughout its height.

Units matter: since the answer is an area, the unit is cm² (not just cm). In CBSE exams, writing 50 cm instead of 50 cm² costs you a mark. Always write the unit explicitly for areas and volumes.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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