Lesson 67 of 100 | Quantitative Aptitude / Plane Mensuration / समतल क्षेत्रमिति
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
- A parallelogram has area 126 cm² and base 18 cm. Find the perpendicular height. Can its perimeter be determined?
- A rhombus has diagonals 10 cm and 24 cm. Find area and perimeter.
- A trapezium has parallel sides 19 m and 11 m, separated perpendicularly by 9 m. Find area.
- A trapezium has area 210 cm², perpendicular height 12 cm and one parallel side 21 cm. Find the other parallel side.
Worked solutions
- Height = 126/18 = 7 cm. The adjacent side is unknown, so perimeter is not determined; height need not equal that side.
- Area = 10 × 24/2 = 120 cm². Side = √(5² + 12²) = 13 cm. Perimeter = 4 × 13 = 52 cm.
- Sum of parallel sides = 19 + 11 = 30 m. Area = 30 × 9/2 = 135 m².
- Let the missing base be b cm. Then 210 = (21 + b) × 12/2 = 6(21 + b). Hence 35 = 21 + b, giving b = 14 cm.