A rectangular solid (a box, or right rectangular prism) has three dimensions — length, width and height — and is built from 6 rectangular faces, 12 edges and 8 vertices, with opposite faces identical. Its volume is simply the product of the three dimensions: V = l × w × h. A cube is the special case where all 12 edges are equal, every face is a square, and the volume collapses to V = s³. Reverse questions are common: given a cube’s volume, take the cube root to recover the edge. Because the figure is not drawn to scale, rely on the stated dimensions, and keep the units consistent before multiplying — mixing centimetres and metres is a classic way to be off by a factor of a thousand.
✅ Solved examples
1. A box measures 4 by 3 by 2. Find its volume.
V = l × w × h = 4 × 3 × 2 = 24.
2. A cube has an edge of 5. Find its volume.
V = s³ = 5³ = 125.
3. A cube has a volume of 64. Find the length of an edge.
s³ = 64 → s = ∛64 = 4.
4. A box has a square base of side 3 and a height of 7. Find its volume.
V = (3 × 3) × 7 = 9 × 7 = 63.
✏️ Practice — try these, take hints as needed
1. A box measures 6 by 2 by 3. Find its volume.
V = l × w × h.
6 × 2 × 3.
Multiply across.
36
2. A cube has an edge of 3. Find its volume.
V = s³.
3 × 3 × 3.
Cube it.
27
3. A cube has a volume of 216. Find the length of an edge.
s³ = 216.
Take the cube root.
6 × 6 × 6 = 216.
6
4. A box has a square base of side 4 and a height of 5. Find its volume.
Base area = 4 × 4.
= 16.
Multiply by height 5.
80
5. A cube has an edge of 10. Find its volume.
V = s³.
10³.
10 × 10 × 10.
1000
📝 Topic test — 8 questions
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