Parallelogram, rhombus and trapezium areas

Learning outcome

Find areas using heights or diagonals, distinguish sides from heights, and assess perimeter data.

Concepts and assumptions

Use exact lengths and consistent units. A parallelogram’s opposite sides are parallel and equal. Moving a triangular end piece to the other end produces a rectangle without changing area. Therefore area = base × perpendicular height. Its perimeter is twice the sum of adjacent side lengths, not twice the sum of base and height.

A rhombus has four equal sides. Its diagonals bisect each other perpendicularly, creating four right triangles with legs equal to half-diagonals. Adding gives rhombus area = diagonal₁ × diagonal₂ ÷ 2. Perimeter is four times a side; diagonals are not boundary sides.

Here, a trapezium has two opposite parallel sides, called bases, while the other two sides are not parallel. Two identical copies form a parallelogram with the same height and base equal to the sum of the parallel sides. Thus trapezium area = (sum of parallel sides) × perpendicular height ÷ 2.

Height means perpendicular separation, never an arbitrary sloping side. Bases and height determine trapezium area, but not generally its perimeter. Extra information, such as equal non-parallel sides, can supply missing lengths.

Worked examples

Example 1 — Parallelogram. A parallelogram has base 18 cm, adjacent side 10 cm and perpendicular height 8 cm to that base. Find area and perimeter.

Area = 18 × 8 = 144 cm². Perimeter = 2(18 + 10) = 56 cm. The 8 cm height measures separation; it does not replace the 10 cm boundary side.

Example 2 — Rhombus. A rhombus has diagonals 16 cm and 30 cm. Find area and perimeter.

Area = 16 × 30/2 = 240 cm². Half-diagonals are 8 cm and 15 cm. A side is the hypotenuse of their right triangle: side = √(8² + 15²) = √289 = 17 cm. Perimeter = 4 × 17 = 68 cm.

Example 3 — Isosceles trapezium. Parallel bases measure 26 cm and 14 cm, with perpendicular separation 8 cm. The shorter base is centred above the longer, making the non-parallel sides equal.

Area = (26 + 14) × 8/2 = 160 cm². Each horizontal end offset = (26 − 14)/2 = 6 cm. Each sloping side = √(6² + 8²) = √100 = 10 cm. Perimeter = 26 + 14 + 10 + 10 = 60 cm.

Common mistakes

Using a sloping side as height; adding diagonals to obtain a rhombus perimeter; averaging non-parallel sides for trapezium area; inferring perimeter from insufficient data.

Practice questions

1. A parallelogram has area 126 cm² and base 18 cm. Find the perpendicular height. Can its perimeter be determined?
2. A rhombus has diagonals 10 cm and 24 cm. Find area and perimeter.
3. A trapezium has parallel sides 19 m and 11 m, separated perpendicularly by 9 m. Find area.
4. A trapezium has area 210 cm², perpendicular height 12 cm and one parallel side 21 cm. Find the other parallel side.

Worked solutions

1. Height = 126/18 = 7 cm. The adjacent side is unknown, so perimeter is not determined; height need not equal that side.
2. Area = 10 × 24/2 = 120 cm². Side = √(5² + 12²) = 13 cm. Perimeter = 4 × 13 = 52 cm.
3. Sum of parallel sides = 19 + 11 = 30 m. Area = 30 × 9/2 = 135 m².
4. Let the missing base be b cm. Then 210 = (21 + b) × 12/2 = 6(21 + b). Hence 35 = 21 + b, giving b = 14 cm.