Volume tells you how much a 3D shape can hold; surface area tells you how much material you’d need to wrap it. Boxes, cans, balls, and cones each have their own formulas, but they all share one logic: build up from the 2D area of the base. Master these four solids and you’ll handle any volume or surface area problem a test can throw at you.
1. Rectangular Boxes (Prisms)
A box is defined by its length (l), width (w), and height (h):
- Volume: V = l × w × h
- Surface area: SA = 2(lw + lh + wh) — the three distinct face-pairs, front/back, top/bottom, left/right, each counted twice
Worked Example: Box
Find the volume and surface area of a box with length 8 cm, width 5 cm, height 4 cm.
- Volume: V = 8 × 5 × 4 = 160 cm³
- Surface area: SA = 2[(8 × 5) + (8 × 4) + (5 × 4)] = 2[40 + 32 + 20] = 2[92] = 184 cm²
2. Right Circular Cylinders
A cylinder has two identical circular bases of radius r, separated by height h:
- Volume: V = π × r² × h (base area × height)
- Surface area: SA = 2πr² + 2πrh — the two circular ends plus the side wall, which unrolls into a rectangle of width 2πr and height h
Worked Example: Cylinder
Find the volume and surface area of a cylinder with radius 3 m and height 7 m (use π ≈ 3.14).
- Volume: V = 3.14 × (3)² × 7 = 3.14 × 9 × 7 = 197.82 m³
- Surface area: SA = 2(3.14)(3)² + 2(3.14)(3)(7) = 56.52 + 131.88 = 188.4 m²
For cylinders, always square the radius before multiplying by height and π. And double-check whether the problem gives you a diameter — halve it first.
3. Spheres
A sphere needs only one measurement — the radius r:
- Volume: V = (4/3) × π × r³
- Surface area: SA = 4 × π × r²
Worked Example: Sphere
Find the volume and surface area of a sphere with radius 6 cm (use π ≈ 3.14).
- Volume: V = (4/3) × 3.14 × (6)³ = (4/3) × 3.14 × 216 = (4/3) × 678.24 = 904.32 cm³
- Surface area: SA = 4 × 3.14 × (6)² = 4 × 3.14 × 36 = 452.16 cm²
4. Right Circular Cones
A cone has a circular base of radius r, a vertical height h, and a slant height s (the diagonal from tip to base edge):
- Volume: V = (1/3) × π × r² × h — exactly one-third of the cylinder with the same base and height
- Surface area: SA = πr² + πrs — the base plus the curved side
Worked Example: Cone
Find the volume and surface area of a cone with radius 4 cm, vertical height 9 cm, and slant height 10 cm (use π ≈ 3.14).
- Volume (needs vertical height): V = (1/3) × 3.14 × (4)² × 9 = (1/3) × 3.14 × 16 × 9 = (1/3) × 452.16 = 150.72 cm³
- Surface area (needs slant height): SA = 3.14 × (4)² + 3.14 × 4 × 10 = 50.24 + 125.6 = 175.84 cm²
3 Mistakes to Avoid
- Mixing up cubic and square units: volume is always cubic (cm³, m³); surface area is always square (cm², m²). Writing the wrong unit is an instant mark off.
- Using diameter instead of radius: every formula here wants r. If the problem gives a diameter, halve it before you do anything else.
- Using slant height for cone volume: volume needs the vertical 90° height (h); only surface area uses the slant height (s). They are not interchangeable.
Volume grows much faster than surface area as a shape gets bigger — double every dimension and volume multiplies by 8 while surface area only quadruples. That’s why elephants overheat less easily than mice, and why it matters in engineering.