Workshop Calculation: Mensuration - Length, Area and Volume of Standard Shapes
Mensuration of standard shapes
This domain tests perimeter, area and volume of rectangles, triangles, circles, trapeziums, cylinders, cones and spheres, with unit consistency. Workshop questions often ask for sheet material area, bar volume or container capacity.
Core formulas
- Rectangle: A = lb, P = 2(l + b); triangle: A = 1/2 bh; trapezium: A = 1/2(a + b)h.
- Circle: circumference = πD = 2πr; area = πr². Use r = D/2.
- Cylinder: volume = πr²h; curved area = 2πrh; total area = 2πr² + 2πrh.
- Cone: volume = 1/3πr²h; slant height l = √(r² + h²); curved area = πrl.
- Sphere: volume = 4/3πr³; surface area = 4πr².
- 1 cm = 10 mm, so 1 cm² = 100 mm² and 1 cm³ = 1000 mm³; 1 litre = 1000 cm³.
Worked example
A 300 mm × 200 mm plate has a 50 mm diameter hole. Plate area = 300 × 200 = 60,000 mm². Hole radius = 25 mm, so hole area = π × 25² = 1,963.5 mm². Net area = 60,000 − 1,963.5 = 58,036.5 mm² = 580.365 cm². If the plate is 1 mm thick, its volume is 58,036.5 mm³ = 58.04 cm³. The reasoning is to subtract the hole from the full plate and convert only after all dimensions are in the same unit.
Traps
- Using diameter instead of radius in πr².
- Mixing mm and cm inside one calculation.
- Adding areas of overlapping shapes instead of subtracting the overlap.
- Forgetting both circular ends on a closed cylinder.
- Using vertical height instead of slant height for a cone’s curved area.