Area of Trapeziums and Rhombi: Formulas

A trapezium has one pair of parallel sides, and a rhombus has four equal sides with diagonals crossing at right angles. Their area formulas look different from the rectangle's, but both are clever shortcuts built on the same base × height idea.

These formulas matter because trapeziums and rhombi appear in tiling, design, and land measurement, where rectangles are too plain to describe the shape. Knowing both formulas — and why each works — prevents the most common area mistakes.

Key Concepts

• Trapezium: the two parallel sides are the bases (a and b); the height h is the perpendicular distance between them.
• Why the trapezium formula works: two identical trapeziums placed head-to-tail form a parallelogram with base (a + b) and height h, so one trapezium has half that area: (a + b)/2 × h.
• Rhombus: all sides equal; the diagonals bisect each other at right angles, cutting the rhombus into four congruent right-angled triangles.
• Why the rhombus formula works: the four right triangles each have legs d1/2 and d2/2, so total area = 4 × (1/2)(d1/2)(d2/2) = (d1 × d2)/2.
• Units: both areas are in square units (cm², m²).

Formulas

• Trapezium: A = (a + b) / 2 × h, where a and b are the parallel sides and h is the perpendicular height between them.
• Rhombus (diagonals): A = (d1 × d2) / 2, where d1 and d2 are the two diagonals.
• Rhombus (base and height): A = base × height, like any parallelogram — use this when the height is known instead of the diagonals.
• Median shortcut: the trapezium's mid-segment m = (a + b)/2, so A = m × h also works.

Worked Example

A rhombus has diagonals 10 cm and 6 cm. A trapezium has parallel sides 12 cm and 8 cm with height 5 cm. Find both areas.

1. Rhombus: A = (10 × 6) / 2 = 60 / 2 = 30 cm².
2. Trapezium: average of bases = (12 + 8) / 2 = 10; A = 10 × 5 = 50 cm².
3. Check the rhombus against base × height thinking: a rhombus with diagonals 10 and 6 has side √(25 + 9) = √34 ≈ 5.83 cm; base × height would give the same 30 cm². Consistent.

Common Misconceptions

• Average the bases — do not just add them. Forgetting the divide-by-2 in the trapezium formula doubles the answer.
• The rhombus formula needs both diagonals. Multiplying one diagonal by a side, or forgetting to halve the product, gives a wrong area.
• Height must be perpendicular in the trapezium. Using a non-parallel leg as the height overestimates the area, exactly as with parallelograms.

Real-World Applications

• Kite making: traditional diamond kites are rhombi, and makers use (d1 × d2)/2 to cut the right amount of paper or fabric.
• Road building: embankment cross-sections are trapeziums; the area formula times the road length gives the volume of earth needed.
• Floor inlays: rhombus and trapezium tiles create striking patterns, and tilers compute areas to order materials.
• Signage: many warning signs use rhombus (diamond) shapes, sized with the diagonal formula.

A trapezium's area is the average of its parallel sides times the perpendicular height, and a rhombus's area is half the product of its diagonals (or base × height, like any parallelogram). Both formulas come from rearranging the shape into rectangles or triangles. Keep the height perpendicular and never skip the divide-by-two, and these areas are always correct.