Volume of three-dimensional figures

Space inside

Volume is the amount of space a solid takes up, measured in cubic units (cm³, m³). A 1 cm³ cube holds 1 mL.

Prisms and cylinders: base × height

For any prism or cylinder:

V = base area × height

Example: Cylinder r = 4 cm, h = 10 cm

V = π(16)(10) = 160π ≈ 503 cm³

Pyramids and cones: one third

A pyramid or cone holds exactly one third of the prism or cylinder with the same base and height.

Example: Square pyramid, base 6 m, height 9 m

V = ⅓ × 36 × 9 = 108 m³

Try it: fill a cone with rice and pour it into a cylinder of the same size. It takes three cones to fill it.

Sphere

V = (4/3)πr³

Example: r = 3 cm → (4/3)π(27) = 36π ≈ 113 cm³

Volume and capacity

Capacity is how much a container holds.

• 1 cm³ = 1 mL
• 1,000 cm³ = 1 L
• 1 m³ = 1,000 L

A fish tank 60 cm × 30 cm × 40 cm = 72,000 cm³ = 72 L.

Finding a missing dimension

Example: A cylinder holds 1,000 cm³ and has r = 5 cm. Height?

1,000 = π(25)h → h = 1,000/(25π) ≈ 12.7 cm

Effect of scaling

Double every dimension → volume × 2³ = 8 times.

Worked examples

Example 1: Rectangular prism 8 × 5 × 3: 120 units³.

Example 2: Cone r = 6, h = 10: ⅓π(36)(10) = 120π ≈ 377 units³.

Example 3: A box holds 2.4 L. Is that 2,400 cm³? Yes.

Common mistakes

• Forgetting the ⅓ for pyramids and cones.
• Using diameter instead of radius.
• Using slant height instead of vertical height.
• Square units for volume.

Try it yourself

1. Cube edge 5 cm
2. Cylinder r = 3, h = 8 (exact)
3. Pyramid with base area 50 m² and height 12 m
4. Sphere r = 6 (exact)
5. How many litres in a 2 m × 1 m × 0.5 m tank?

Answers: 1) 125 cm³ 2) 72π 3) 200 m³ 4) 288π 5) 1,000 L

Quick recap

• Volume = space inside, in cubic units.
• Prisms and cylinders: base area × height; pyramids and cones: one third of that.
• 1 cm³ = 1 mL connects volume and capacity.