Volume & Surface Area – Quick Reference Cards

Cube

Volume

V = a^3

Surface Area

SA = 6a^2

Cuboid

Volume

V = l \cdot w \cdot h

Surface Area

SA = 2(lw+lh+wh)

Cylinder

Volume

V = \pi r^2 h

Surface Area

SA = 2\pi r^2 + 2\pi r h

Cone

Volume

V = \frac{1}{3}\pi r^2 h

Surface Area

SA = \pi r^2 + \pi r l

Sphere

Volume

V = \frac{4}{3}\pi r^3

Surface Area

SA = 4\pi r^2

Square Pyramid

Volume

V = \frac{1}{3} a^2 h

Surface Area

SA = a^2 + 2al

Real-World Applications

Real world: packaging design — how much cardboard for a cereal box?

A cereal box (cuboid) is 30 cm × 8 cm × 20 cm. Surface area = amount of cardboard needed:

$SA = 2(30 \times 8 + 30 \times 20 + 8 \times 20) = 2(240 + 600 + 160) = 2{,}000 \text{ cm}^2$

Volume = how much cereal fits: $V = 30 \times 8 \times 20 = 4{,}800 \text{ cm}^3$ .

Real world: architecture — concrete for cylindrical pillars

A building needs 12 concrete pillars, each a cylinder with radius 0.3 m and height 4 m.

Volume of one pillar: $V = \pi \times 0.09 \times 4 \approx 1.13 \text{ m}^3$

Total concrete: $12 \times 1.13 \approx 13.6 \text{ m}^3$ — about 14 ready-mix lorry loads.

Real world: manufacturing — paint for spherical gas tanks

A storage facility has 6 spherical tanks, each with radius 5 m. Total surface area to be painted:

$SA_{\text{one}} = 4\pi \times 25 = 100\pi \approx 314 \text{ m}^2$

Total: $6 \times 314 = 1{,}884 \text{ m}^2$ — the quantity needed when ordering industrial paint.

Tip

Choose the right formula: Use volume when filling, pouring, or storing something inside a solid. Use surface area when covering, painting, or wrapping the outside. The shape names in the cards above link directly to everyday objects.