Mensuration

Summary: Covers perimeter, area, surface area, and volume formulas for 2D and 3D shapes, including circles, cylinders, pyramids, cones, spheres, and compound shapes. Includes unit conversions, nets, and plan/elevation drawings. Essential for IGCSE Mathematics 0580 — questions often combine mensuration with algebra or trigonometry. Tags: igcse mathematics mensuration measurement Created: 2026-07-16 Last Updated: 2026-07-16

1. Perimeter and Area of 2D Shapes

Rectangle: Perimeter = 2(l + w), Area = l × w

Triangle: Area = 1/2 × base × perpendicular height

For any triangle (Extended): Area = 1/2 ab sin C; Heron’s formula: Area = √[s(s − a)(s − b)(s − c)] where s = (a + b + c)/2.

Parallelogram: Area = base × perpendicular height (NOT slant height)

Trapezium: Area = 1/2 (a + b) × h, where a and b are the parallel sides, h is the perpendicular distance between them.

Circle:

• Circumference: C = 2πr = πd
• Area: A = πr²

Example: Find the area of a trapezium with parallel sides 8 cm and 12 cm, height 5 cm. Area = 1/2 (8 + 12) × 5 = 50 cm².

Example: A circle has radius 7 cm. Find circumference and area (π ≈ 22/7). C = 2 × (22/7) × 7 = 44 cm. A = (22/7) × 49 = 154 cm².

Arc length and sector area:

• Arc length = (θ/360) × 2πr
• Sector area = (θ/360) × πr²
• θ is the angle at the centre, in degrees.

Example: Find the arc length and sector area of a circle radius 10 cm, sector angle 72°. Arc length = (72/360) × 2π × 10 = 4π cm (≈ 12.57 cm). Sector area = (72/360) × π × 100 = 20π cm² (≈ 62.83 cm²).

2. Surface Area and Volume of 3D Shapes

Cube / Cuboid:

• Volume = length × width × height
• Surface area = 2(lw + lh + wh)

Prism (any shape with a uniform cross-section):

• Volume = area of cross-section × length
• Surface area = sum of areas of all faces (including the two ends)

Cylinder:

• Volume = πr²h
• Curved surface area = 2πrh
• Total surface area = 2πrh + 2πr²

Example: A cylinder has radius 3 cm and height 10 cm. Find volume and total surface area. V = π × 9 × 10 = 90π cm³ (≈ 282.7 cm³). Total SA = 2π × 3 × 10 + 2π × 9 = 60π + 18π = 78π cm² (≈ 245.0 cm²).

Pyramid:

• Volume = 1/3 × area of base × perpendicular height
• Surface area = area of base + sum of areas of triangular faces

Cone:

• Volume = 1/3 πr²h
• Curved surface area = πrl, where l = √(r² + h²) (slant height)
• Total surface area = πrl + πr²

Example: A cone has radius 6 cm and height 8 cm. Find (a) slant height, (b) volume, (c) curved surface area. (a) l = √(6² + 8²) = √100 = 10 cm. (b) V = 1/3 × π × 36 × 8 = 96π cm³ (≈ 301.6 cm³). (c) Curved SA = π × 6 × 10 = 60π cm² (≈ 188.5 cm²).

Sphere:

• Volume = 4/3 πr³
• Surface area = 4πr²

Hemisphere:

• Volume = 2/3 πr³
• Curved surface area = 2πr²
• Total surface area = 2πr² + πr² = 3πr² (curved surface + circular base)

Example: A sphere has radius 5 cm. Find volume and surface area. V = 4/3 × π × 125 = 500π/3 cm³ (≈ 523.6 cm³). SA = 4 × π × 25 = 100π cm² (≈ 314.2 cm²).

Example: A solid consists of a cone on top of a hemisphere of radius 3 cm. The cone has height 4 cm. Find the total volume. Hemisphere V = 2/3 × π × 27 = 18π. Cone V = 1/3 × π × 9 × 4 = 12π. Total = 30π cm³ (≈ 94.2 cm³).

3. Units

Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm

Area (square units):

• 1 m² = 10,000 cm² (NOT 100 cm² — square the conversion factor: 100² = 10,000)
• 1 hectare = 10,000 m²
• 1 km² = 1,000,000 m²

Volume / Capacity (cubic units):

• 1 m³ = 1,000,000 cm³ (cube the conversion factor: 100³ = 1,000,000)
• 1 litre = 1000 ml = 1000 cm³
• 1 m³ = 1000 litres

Memory aid:

• Length: ×1000 (km→m), ×100 (m→cm), ×10 (cm→mm)
• Area: square the length factors (×100² = ×10,000 for m²→cm²)
• Volume: cube the length factors (×100³ = ×1,000,000 for m³→cm³)

Example: Convert 5 m² to cm². 5 × 10,000 = 50,000 cm².

Example: Express 450 ml in cm³. 1 ml = 1 cm³, so 450 ml = 450 cm³.

Example: A water tank is a cuboid 2 m × 1.5 m × 0.8 m. How many litres when full? Volume = 2 × 1.5 × 0.8 = 2.4 m³ = 2.4 × 1000 = 2400 litres.

4. Nets of 3D Shapes

A net is a 2D pattern that folds to form a 3D shape.

• Cube: 11 possible nets — a cross of 6 squares. All faces must connect correctly when folded.
• Cuboid: 3 pairs of congruent rectangles.
• Cylinder: A rectangle (curved surface) plus two circles (ends).
• Cone: A sector of a circle (curved surface) plus one circle (base).
• Pyramid: Base polygon plus triangles for each side.

5. Plan and Elevation Drawings

Plan: View from directly above (bird’s-eye view).

Front elevation: View from the front. Side elevation: View from the side.

These are 2D projections of a 3D shape.

Example: Describe the plan and front elevation of a cylinder standing upright. Plan: a circle. Front elevation: a rectangle (width = diameter, height = cylinder height).

Sources

• BBC Bitesize GCSE Mathematics — Perimeter, Area, and Volume, BBC (free educational resource)
• OpenStax Math — Measurement and Geometry, Rice University (free, CC BY 4.0)
• Cambridge IGCSE Mathematics 0580 — Mensuration, Cambridge Assessment International Education
• CK-12 Mathematics — Measurement, CK-12 Foundation (free, CC BY-NC 3.0)

Related Notes

• Geometry — Angle facts, polygons, symmetry used in mensuration problems
• Algebra and Graphs — Rearranging formulas, solving for missing dimensions
• Trigonometry — Area = 1/2 ab sin C, Pythagoras, 3D trigonometry
• Number — Upper and lower bounds applied to measurements
• IGCSE-Maths-Index — Full IGCSE Mathematics index

Common Misconceptions