Surface area and volume of cubes and cuboids

Learning outcome

Calculate surface area and volume of cubes and cuboids, choosing the correct faces and dimensions for a closed solid or an open container.

Concepts and assumptions

Surface area measures boundaries in square units; volume measures space in cubic units. Treat dimensions as exact. Containers have negligible wall thickness, and capacity uses internal dimensions.

For a cuboid, choose a rectangular base l by b and perpendicular height h. Each horizontal layer has area l × b; stacking layers through height h gives V = l × b × h.

Opposite faces form three equal pairs, so total surface area, TSA = 2 × (l × b + b × h + h × l). Lateral surface area, LSA = 2 × h × (l + b), includes four walls but excludes both designated bases.

For a cube with edge a, V = a³, TSA = 6 × a² and LSA = 4 × a².

A top-open cuboidal container has one base and four walls: area = l × b + 2 × h × (l + b). This counts one specified surface of each wall, not both inside and outside. Exposed area counts only uncovered or requested surfaces.

Use 1 L = 1000 cm³. Convert lengths to compatible units before multiplying. No rounding is needed.

Worked examples

Example 1 — Cube. A solid cube has edge 5 cm. Each face has area 5 × 5 = 25 cm². LSA = 4 × 25 = 100 cm²; TSA = 6 × 25 = 150 cm². Volume = 5 × 5 × 5 = 125 cm³.

Example 2 — Cuboid. A solid cuboid is 18 cm long, 11 cm broad and 7 cm high. Face areas are 198, 77 and 126 cm². TSA = 2 × (198 + 77 + 126) = 802 cm². Volume = 18 × 11 × 7 = 1386 cm³.

Example 3 — Open tank. A top-open tank has internal dimensions 60 cm × 35 cm × 40 cm high. Coat only its inner base and four inner walls. Area = 60 × 35 + 2 × 40 × (60 + 35) = 2100 + 7600 = 9700 cm². Brimful capacity = 60 × 35 × 40 = 84000 cm³ = 84 L.

Common mistakes

Do not add a nonexistent top, count unrequested surfaces, or use external dimensions for internal capacity. Area and volume require different units and different conversion factors.

Practice questions

1. A cube has edge 7 cm. Find its lateral area, total area and volume.
2. A cuboid measures 16 cm × 9 cm × 5 cm. Find its total area and volume.
3. A top-open tank has internal length 50 cm, breadth 32 cm and height 25 cm. Find its inner base-and-wall area and capacity.
4. A cuboid has a 12 cm × 8 cm base and volume 960 cm³. Find its perpendicular height and lateral area.

Worked solutions

1. LSA = 4 × 7² = 196 cm²; TSA = 6 × 7² = 294 cm². Volume = 7³ = 343 cm³.
2. TSA = 2 × (144 + 45 + 80) = 538 cm². Volume = 16 × 9 × 5 = 720 cm³.
3. Area = 50 × 32 + 2 × 25 × (50 + 32) = 1600 + 4100 = 5700 cm². Capacity = 50 × 32 × 25 = 40000 cm³ = 40 L.
4. Base area = 12 × 8 = 96 cm². Height = 960 ÷ 96 = 10 cm. LSA = 2 × 10 × (12 + 8) = 400 cm².