Volume measures how much space a solid encloses, always in cubic units. Three formulas cover the entire GRE: box V = l × w × h, cube V = s³, and cylinder V = πr²h. Two non-obvious ideas separate strong test-takers here. First, scaling: if you multiply every linear dimension by k, the volume multiplies by k³ — doubling a cube’s edge makes it eight times as large, not twice. Second, "capacity" and "how many small boxes fit" questions are volume-division problems, but they only work cleanly when the small dimensions divide the large ones evenly. Keep every measurement in one unit, and note that 1 litre = 1000 cubic centimetres if a problem crosses between capacity and length.
✅ Solved examples
1. A box measures 10 by 5 by 4. Find its volume.
V = 10 × 5 × 4 = 200.
2. A cube’s edge is doubled from 2 to 4. By what factor does its volume increase?
Old volume = 2³ = 8; new = 4³ = 64. Factor = 64 / 8 = 8 (volume scales as the cube of the linear factor).
3. A rectangular water tank measures 2 m by 3 m by 5 m. Find its volume in cubic metres.
V = 2 × 3 × 5 = 30 cubic metres.
4. A cylinder has a radius of 10 and a height of 7. Find its volume.
V = πr²h = π × 100 × 7 = 700π.
✏️ Practice — try these, take hints as needed
1. A box measures 8 by 5 by 2. Find its volume.
V = l × w × h.
8 × 5 × 2.
Multiply across.
80
2. A cube has an edge of 4. Find its volume.
V = s³.
4³.
4 × 4 × 4.
64
3. A cube’s edge is tripled. By what factor does its volume increase?
Volume scales as the cube of the linear factor.
3³.
3 × 3 × 3.
27 times
4. A cylinder has a radius of 6 and a height of 5. Find its volume.
V = πr²h.
r² = 36.
π × 36 × 5.
180π
5. A box has a volume of 120 and a base measuring 6 by 5. Find its height.
Base area = 30.
120 = 30 × h.
Divide by 30.
4
📝 Topic test — 8 questions
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