What Is a Trapezium?
In UK / Indian usage (and this article): a trapezium is a quadrilateral with exactly one pair of parallel sides.
In US usage, the same shape is called a trapezoid — and the word trapezium means no parallel sides. This terminology conflict has historical roots and persists today.
For clarity, this article uses UK terminology. If you're reading US-published materials, trapezoid means what we call trapezium.
The parallel sides are the bases (usually $a$ and $b$). The non-parallel sides are the legs. The perpendicular distance between the bases is the height $h$.
Types of Trapezium
Scalene Trapezium
All four sides have different lengths. The most general type.
Isosceles Trapezium
The two non-parallel sides (legs) are equal in length. Has an axis of symmetry perpendicular to the bases. Both base angles at each base are equal.
Right Trapezium
Has two right angles (one of the legs is perpendicular to the bases).
Area Formula
$$A = \frac{1}{2}(a + b) h$$
where $a$ and $b$ are the lengths of the two parallel sides (bases) and $h$ is the perpendicular height between them.
Interpretation: The area is the average of the two bases multiplied by the height.
Derivation: A trapezium can be divided into a rectangle plus two triangles, or two triangles back-to-back. Either way, the algebra yields $\tfrac{1}{2}(a + b) h$.
Perimeter Formula
$$P = a + b + c + d$$
where $a, b$ are the bases and $c, d$ are the legs. Just add all four side lengths.
Three Worked Examples — Quick, Standard, Stretch
Quick — Area
A trapezium has parallel sides 8 cm and 12 cm, and height 5 cm. Find its area.
$$A = \tfrac{1}{2}(8 + 12)(5) = \tfrac{1}{2}(20)(5) = 50 \text{ cm}^2$$
Standard — Find Missing Base
A trapezium has area $40$ m², one base $6$ m, and height $4$ m. Find the other base.
$$40 = \tfrac{1}{2}(6 + b)(4) = 2(6 + b)$$
$20 = 6 + b$, so $b = 14$ m.
Stretch — Isosceles Trapezium Properties
An isosceles trapezium has bases $10$ cm and $4$ cm, with legs of length $5$ cm. Find its height and area.
The horizontal "overhang" on each side (between the projection of the shorter base onto the longer base and the longer base's edges) is $(10 - 4)/2 = 3$ cm. Using the Pythagorean theorem on the right triangle formed by half the difference, the height, and the leg:
$$h^2 + 3^2 = 5^2 \implies h^2 = 16 \implies h = 4 \text{ cm}$$
Area: $A = \tfrac{1}{2}(10 + 4)(4) = \tfrac{1}{2}(14)(4) = 28$ cm².
Properties of an Isosceles Trapezium
Both legs equal in length.
Both pairs of base angles equal — angles at the longer base are equal to each other; angles at the shorter base are equal to each other.
Diagonals are equal in length.
Has one axis of symmetry — perpendicular to the bases, through their midpoints.
Can be inscribed in a circle (cyclic quadrilateral) — opposite angles sum to $180°$.
Why Does the Trapezium Matter? (The Real-World GROUND)
"A trapezium is what you get when you slice a triangle off the top of a triangle."
Trapeziums appear in:
Engineering and architecture. Roof trusses, dam cross-sections, retaining wall profiles — all commonly trapezoidal.
Drainage and canals. A canal's cross-section is typically trapezoidal (wider at the top than the bottom for stability).
Numerical integration. The Trapezoidal rule approximates $\int_a^b f(x) dx$ by treating thin strips under the curve as trapeziums.
Currency and lottery design. Many bank notes and lottery tickets feature trapezoidal anti-counterfeiting elements.
Music theory. The boundary of a tuning system's "trapezoid" of consonant intervals.
The systematic study of the trapezium dates to Euclid's Elements (Book I), where quadrilaterals are classified by their parallel-side properties.
Learn more: Parallelogram
A Worked Example
Find the area of a trapezium with bases $7$ cm and $11$ cm, and slant height (leg length) $5$ cm.
The intuitive (wrong) approach. A student uses the slant height $5$ as $h$ in the formula:
$$A \stackrel{?}{=} \tfrac{1}{2}(7 + 11)(5) = 45 \text{ cm}^2$$
Why it fails. The "height" in the area formula is the perpendicular distance between the parallel sides — not the slant length of a leg. Using the leg length overestimates the area when the trapezium is slanted.
The correct method. If only the slant length is given, you need more information (e.g., whether it's isosceles, or the base difference) to compute the perpendicular height. The slant length alone isn't enough — and substituting it into the formula gives a wrong answer.
What Are the Most Common Mistakes With the Trapezium?
Mistake 1: Using the leg length as the height
The fix: Height = perpendicular distance between the parallel sides. The legs are usually slanted; their length is not the height.
Mistake 2: Forgetting the factor of $\tfrac{1}{2}$
The fix: Area is $\tfrac{1}{2}(a + b)h$, not $(a + b)h$. Without the half, you're computing the area of the rectangle of dimensions $(a+b) \times h$, which is twice the trapezium's area.
Mistake 3: Adding both bases without averaging
The fix: The formula uses the average of the bases (which is $(a+b)/2$), times the height. Some students multiply by one base only.
Key Takeaways
A trapezium is a quadrilateral with one pair of parallel sides (the bases).
Area formula: $A = \tfrac{1}{2}(a + b) h$ — average of the bases times the height.
Types: scalene (all sides different), isosceles (equal legs), right (one leg perpendicular to bases).
Height is perpendicular — not the leg length.
UK trapezium = US trapezoid — same shape, different name.
A Practical Next Step
Try these three before moving on to parallelograms.
Find the area of a trapezium with bases $12$ cm and $8$ cm, height $5$ cm.
An isosceles trapezium has bases $14$ and $6$, legs of length $5$. Find its height.
A trapezium has area $60$ m², bases $7$ m and $13$ m. Find the height.
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