Volume – How Much Space a Solid Occupies

Volume measures the amount of three-dimensional space that a solid occupies. It tells you how much liquid a container holds, how much concrete to order for a foundation, or how much air is inside a room.

According to a famous story, the Greek mathematician Archimedes discovered a way to measure the volume of irregular objects while stepping into a public bath and noticing the water level rise around him; realising that the water displaced must exactly equal his own submerged volume, he is said to have run through the streets shouting “Eureka!” (“I have found it!”) — legend has it he then used the same principle to check whether a king's crown was pure gold by comparing its volume, via water displacement, to an equal weight of pure gold. Precise volume calculations remain essential in modern logistics and medicine alike: shipping companies calculate the exact volume of every container to plan cargo loads efficiently, and pharmacists must measure liquid medicine volumes with great accuracy, since even a small error in dosage volume can be dangerous.

What Is Volume?

Volume is measured in cubic units: cm³, m³, litres (1 litre = 1000 cm³). It counts how many unit cubes fit inside a solid shape.

Volume as Stacked Areas

Think of a solid as a stack of identical flat slices. If each slice has a cross-sectional area, and you stack a certain height of them, the total volume is simply that area multiplied by the height.

Each of the 5 horizontal layers has exactly the same cross-sectional area, l × w. Stacking 5 of them gives a total volume of (l × w) × 5 — which is exactly the formula V = l × w × h, with h simply counting how many layers are stacked. This “area × height” idea works for any prism, not just rectangular ones: a cylinder is a stack of circular slices, so its volume is also cross-sectional area (πr²) × height, giving V = πr²h.

Volume Formulas

Solid	Formula	Variables
Cube	V = s³	s = side length
Cuboid	V = l × w × h	l, w, h = dimensions
Cylinder	V = πr²h	r = radius, h = height
Cone	V = ⅓πr²h	r = radius, h = perpendicular height
Sphere	V = &frac43;πr³	r = radius
Triangular prism	V = ½bhl	b, h = triangle base/height, l = length
Any prism	V = cross-section area × length
Any pyramid	V = ⅓ × base area × height

Worked Examples

Find the volume of a cuboid 8 cm × 5 cm × 3 cm.

V = 8 × 5 × 3 = 120 cm³.

Find the volume of a cylinder with radius 4 cm and height 9 cm.

V = π(16)(9) = 144π ≈ 452.39 cm³.

Find the volume of a cone with radius 3 cm and height 7 cm.

V = ⅓π(9)(7) = 21π ≈ 65.97 cm³.

Find the volume of a sphere with radius 6 cm.

V = &frac43;π(216) = 288π ≈ 904.78 cm³.

Key Takeaways

Volume is in cubic units (cm³, m³).
Prisms: V = cross-section area × length. Pyramids/cones: V = ⅓ × base area × height.
Cylinder: πr²h. Sphere: &frac43;πr³. Cone: ⅓πr²h.
1 litre = 1000 cm³; 1 m³ = 1000 litres.

Practice: Volume Calculations

Find the Volume