Volume is the amount of space a 3D solid takes up — how much water a tank holds, how much concrete fills a mould — measured in cubic units (cm³, m³). Surface area is the total area of all its outside faces — how much wrapping paper covers a box, how much paint coats it — measured in square units (cm²).
Prisms — the big idea
A prism is a solid with the same cross-section all the way through — like a loaf of bread where every slice is identical, or a Toblerone box. Because every slice is the same, its volume is just that slice's area, stretched along the length:
- Cuboid (a box): its cross-section is a rectangle, so V = length × width × height.
- Cylinder (a circular prism, like a tin): its cross-section is a circle, so V = πr²h.
- Triangular prism (a tent, a Toblerone): V = (½ × base × height) × length.
Cylinder r = 3, h = 10 (π = 3.14): V = 3.14 × 3² × 10 = 3.14 × 9 × 10 = 282.6 cm³.
Surface area
Surface area ignores the inside entirely — you add up the area of every face, as if unfolding the solid flat like a cardboard net. A cube has 6 identical square faces, so its surface area is 6 × (side)². A cuboid has three pairs of matching rectangles.
| Solid | Volume | Cross-section |
|---|---|---|
| Cuboid | l × w × h | rectangle |
| Cube | side³ | square |
| Cylinder | πr²h | circle |
| Triangular prism | (½ × b × h) × length | triangle |
Memory hook: for any prism, find the end slice's area, then push it through the length. Volume fills the inside (cm³); surface area wraps the outside (cm²).