Practice Maths

Volume

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Key Terms

Volume
The amount of 3D space enclosed by a solid; measured in cubic units (mm³, cm³, m³).
Prism/Cylinder volume
V = base area × height (base area for a cylinder is πr²).
Pyramid/Cone volume
V = ⅓ × base area × height.
Sphere volume
V = &frac43;πr³.
Composite solid
Identify and split into recognisable parts; add or subtract volumes as appropriate.
Capacity
1 mL = 1 cm³; 1 L = 1 000 mL = 1 000 cm³; 1 m³ = 1 000 L.

Volume is the amount of three-dimensional space a solid occupies. It is measured in cubic units (cm³, m³, etc.). Volume is closely related to capacity — the amount a container can hold.

Volume Formulas

SolidFormulaNotes
Prism (any shape)V = AhA = cross-sectional area, h = height
Rectangular prismV = lwhSpecial case of prism
CylinderV = πr²hSpecial case of prism (A = πr²)
Pyramid (any base)V = ⅓AhA = base area, h = perpendicular height
ConeV = ⅓πr²hSpecial case of pyramid
SphereV = &frac43;πr³—
HemisphereV = ⅔πr³Half of sphere

Capacity Conversions

1 cm³ = 1 mL1000 cm³ = 1 L1 m³ = 1000 L = 1 kL
Rectangular Prism l h w V = lwh Cylinder r h V = πr²h Pyramid h V = ⅓Ah Cone r h V = ⅓πr²h Sphere r V = &frac43;πr³

Worked Example 1 — Volume of a cylinder

Find the volume of a cylinder with r = 3 cm and h = 10 cm. Also find its capacity in litres.

V = πr²h = π × 9 × 10 = 90π ≈ 282.74 cm³

Capacity: 282.74 cm³ = 282.74 mL = 0.283 L

Worked Example 2 — Volume of a composite solid

Find the total volume of a cylinder (r = 4, h = 8) with a cone on top (r = 4, h = 3).

Vcylinder = πr²h = π(16)(8) = 128π

Vcone = ⅓πr²h = ⅓π(16)(3) = 16π

Total V = 128π + 16π = 144π ≈ 452.39 cm³

Hot Tip: A pyramid or cone has exactly ⅓ the volume of the prism or cylinder with the same base and height. The ⅓ factor arises because the shape “tapers” to a point. For capacity conversions, remember: 1 cm³ = 1 mL and 1 m³ = 1 kL.

Full Lesson

Volume measures how much 3D space a solid fills. The general principle for all prisms and cylinders is: V = cross-sectional area × height. For tapered solids (pyramids, cones), multiply by ⅓.

Prisms and Cylinders

A prism is a solid with a uniform cross-section. The cross-section is the same shape no matter where you slice it parallel to the base. A cylinder is a circular prism. For both: V = Ah, where A is the base area and h is the height (or length).

Pyramids and Cones

Pyramids and cones taper from a base to a point (apex). They hold exactly one-third the volume of the matching prism/cylinder. This factor of ⅓ cannot be simplified away — it is built into the formula.

The Sphere

V = &frac43;πr³. Note the r is cubed. Volume scales with the cube of the radius, so doubling the radius gives 8 times the volume.

Volume to Capacity

This is a critical connection in practical problems:

  • 1 cm³ = 1 mL
  • 1000 cm³ = 1 L
  • 1 m³ = 1 000 000 cm³ = 1 000 000 mL = 1000 L = 1 kL

Guide Example — Volume of a cone

Find the volume of a cone with r = 6 cm and h = 7 cm.

V = ⅓πr²h = ⅓ × π × 36 × 7 = ⅓ × 252π = 84π ≈ 263.89 cm³

Guide Example — Composite solid volume

A water trough is a triangular prism (triangle base: base 60 cm, height 40 cm; length 120 cm). Find the volume in litres.

Cross-section (triangle) area = ½ × 60 × 40 = 1200 cm²

V = Ah = 1200 × 120 = 144 000 cm³

Capacity = 144 000 mL = 144 L

Tip: When calculating time to fill a container, convert all volumes to the same unit as the flow rate. For example, if flow is in L/min and volume is in m³, convert m³ to L first (multiply by 1000).

Mastery Practice

See Answers ➔

  1. Q1 — Volume of rectangular prisms

    Fluency

    (a) Find the volume of a rectangular prism with dimensions 8 cm × 5 cm × 3 cm.

    (b) Find the volume of a rectangular prism with dimensions 12 cm × 9 cm × 4.5 cm.

  2. Q2 — Volume of cylinders

    Fluency

    (a) Find the volume of a cylinder with r = 4 cm and h = 10 cm.

    (b) Find the volume of a cylinder with r = 6.5 cm and h = 8 cm.

  3. Q3 — Volume of pyramids

    Fluency

    (a) A square pyramid has base 6 cm × 6 cm and perpendicular height 10 cm. Find its volume.

    (b) A rectangular pyramid has base 8 cm × 5 cm and height 12 cm. Find its volume.

  4. Q4 — Volume of cones

    Fluency

    (a) A cone has r = 5 cm and h = 9 cm. Find the volume.

    (b) A cone has r = 8 cm and h = 15 cm. Find the volume.

  5. Q5 — Volume of spheres

    Fluency

    (a) Find the volume of a sphere with r = 6 cm.

    (b) Find the volume of a sphere with d = 10 m.

  6. Q6 — Volume to capacity conversions

    Understanding

    (a) Convert 4500 cm³ to mL and L.

    (b) Convert 2.8 m³ to kL.

    (c) Convert 350 L to cm³.

  7. Q7 — Rectangular swimming pool

    Understanding

    A rectangular swimming pool is 15 m × 8 m × 1.5 m deep.

    (a) Find the volume in m³.

    (b) Find the capacity in kL.

    (c) What is the depth of water when the pool is 3/4 full?

  8. Q8 — Grain silo: cylinder with conical top

    Understanding

    A grain silo consists of a cylinder (r = 3 m, h = 8 m) topped by a cone (r = 3 m, h = 2 m).

    (a) Find the volume of the cylindrical section.

    (b) Find the volume of the conical section.

    (c) Find the total volume.

    (d) Express the total capacity in kL.

  9. Q9 — Ice cream cone with hemisphere

    Problem Solving

    An ice cream cone has r = 3.5 cm and h = 10 cm. It is topped with a hemisphere of ice cream with r = 3.5 cm.

    (a) Find the volume of the cone.

    (b) Find the volume of the hemisphere.

    (c) Find the total volume.

    (d) If one serving = 80 mL, is one ice cream’s total volume more or less than one serving?

  10. Q10 — Water tank filling time and cost

    Problem Solving

    A cylindrical water tank has r = 1.2 m and h = 2.5 m. Water is pumped in at 0.6 m³ per minute. The tank is currently empty.

    (a) Find the total capacity in kL.

    (b) Find the time to fill the tank to 90% capacity (answer in minutes, then in hours and minutes).

    (c) If each kL of water costs $2.80, find the total cost to fill the tank completely.