Mathematics · Lesson 18

Mensuration II: volumes & surface areas of solids

JAMBWAECNECO

What the syllabus asks

JAMB: lengths, areas and volumes of solid shapes (cube, cuboid, cylinder, cone, pyramid, sphere, frustum), surface areas

WAEC: mensuration of solids: cube, cuboid, prism, cylinder, cone, pyramid, frustum, sphere and hemisphere (volume, curved and total surface area), composite solids, capacity, similar solids

NECO: mensuration of solids: cube, cuboid, prism, cylinder, cone, pyramid, frustum, sphere and hemisphere (volume, curved and total surface area), composite solids, capacity, similar solids

What you will learn

Calculate volumes and surface areas of cubes, cuboids and prisms.
Calculate volume, curved and total surface areas of cylinders, including hollow pipes.
Calculate volume and surface area of cones and pyramids, using slant height.
Calculate volume and surface area of spheres and hemispheres.
Find volumes of frustums and composite solids.
Convert between cm³, m³ and litres, and solve capacity and recasting problems.
Use the ratio of volumes of similar solids.

Key points

Volume in cubic units, area in square units; 1 litre = 1000 cm³; 1 m³ = 1000 litres.
Cuboid V = lbh, area 2(lb + bh + lh); cube V = a³, area 6a².
Any prism: V = cross-section × length.
Cylinder: V = πr²h; curved 2πrh; closed total 2πr(h + r); pipe V = πh(R² − r²).
Cone: l² = r² + h²; V = ⅓πr²h; curved πrl.
Pyramid: V = ⅓ × base area × height; faces need their own slant height.
Sphere: V = (4/3)πr³, A = 4πr²; solid hemisphere V = (2/3)πr³, A = 3πr².
Frustum = big cone − small cone (similar), or ⅓πh(R² + Rr + r²).
Composite: add parts, omit hidden faces; recasting conserves volume.
Similar solids: areas k², volumes k³.

Start here: Solids, faces and units

Every container you use is a solid: a tin of tomato paste (cylinder), a carton of juice (cuboid), a funnel (cone), a football (sphere). Mensuration of solids answers two practical questions: how much does it hold? (volume) and how much material covers it? (surface area).

Quantity	Measures	Units
Volume	space inside	cm³, m³ (cubic units)
Surface area	total area of all the outside faces	cm², m² (square units)
Capacity	how much liquid it holds	ml, litres

Conversions: 1 cm³ = 1 ml; 1 litre = 1000 cm³; 1 m³ = 1,000,000 cm³ = 1000 litres.

Be careful converting cubic units: 1 m = 100 cm, but 1 m³ = 100 × 100 × 100 = 1,000,000 cm³, not 100 cm³.

Euler's rule links faces, vertices and edges of any solid with flat faces: F + V − E = 2. A cuboid has 6 faces, 8 vertices and 12 edges: 6 + 8 − 12 = 2 ✓.

Worked example 18.1

A water tank holds 2.5 m³. How many litres is this, and how many 20-litre jerrycans will fill it?

1 m³ = 1000 litres, so 2.5 m³ = 2500 litres.
2500 ÷ 20 = 125 jerrycans.

2500 litres; 125 jerrycans

⚠️ Common mistake: mixing units, e.g. a length in metres with another in centimetres. Convert everything to one unit first.

Try it yourself. Convert 4500 cm³ to litres.

Show solution

4500 ÷ 1000 = 4.5 litres.

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Practice questions

1. Find the volume of a cuboid 12 cm × 5 cm × 4 cm and its surface area.

Show worked answer

240 cm³; 2(60 + 20 + 48) = 256 cm².

2. A cylinder has r = 3.5 cm and h = 8 cm. Find its volume (π = 22/7).

Show worked answer

(22/7) × 12.25 × 8 = 308 cm³.

3. A cone has r = 5 cm, h = 12 cm. Find l and the curved surface area in terms of π.

Show worked answer

l = 13; 65π cm².

4. Find the volume of a sphere of diameter 6 cm in terms of π.

Show worked answer

r = 3; 36π cm³.

5. A pyramid with square base 10 cm has volume 400 cm³. Find its height.

Show worked answer

⅓ × 100 × h = 400 → h = 12 cm.

Key words

Solid: A three-dimensional shape with length, breadth and height.
Face: A flat surface of a solid.
Edge: A line where two faces meet.
Vertex: A corner where edges meet.
Prism: A solid with the same cross-section all along its length.
Cross-section: The shape you see when a solid is cut straight across.
Cylinder: A prism with a circular cross-section.
Cone: A solid with a circular base tapering to a point (apex).
Pyramid: A solid with a polygon base and triangular faces meeting at an apex.
Slant height: The distance from the apex down the sloping surface to the edge of the base.
Frustum: The part of a cone or pyramid left when the top is cut off parallel to the base.
Sphere: A perfectly round solid; every surface point is the same distance (r) from the centre.

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