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.
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 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 Questions
- Find the volume of a cube with side 7 cm.
- A cylindrical tank has radius 2 m and height 5 m. Find its volume in m³ and capacity in litres.
- A cone has base radius 6 cm and volume 120π cm³. Find its height.
- Find the volume of a sphere with diameter 12 cm. Leave in terms of π.
- A triangular prism has a right-angled cross section with legs 6 cm and 8 cm, and length 15 cm. Find its volume.