Volume — sphere, cylinder, cube, cone
Key facts
- Volume is measured in cubic units: cm^3, m^3, and capacity-linked units such as litres after conversion.
- A cube of side a has volume a^3, so doubling every edge makes the volume 8 times.
- A cylinder of radius r and height h has volume pi r^2 h; always halve the diameter before using the formula.
- A cone with the same base radius and vertical height as a cylinder has one-third of that cylinder's volume: (1/3) pi r^2 h.
- A sphere of radius r has volume (4/3) pi r^3, so radius errors grow cubically in the answer.
Key Points at a Glance
- 1
The official CET Graduation syllabus asks for volume of spheres, cylinders, cubes, and cones; this topic should stay centred on those four solids.
- 2
Volume is measured in cubic units: cm^3, m^3, and capacity-linked units such as litres after conversion.
- 3
A cube of side a has volume a^3, so doubling every edge makes the volume 8 times.
- 4
A cylinder of radius r and height h has volume pi r^2 h; always halve the diameter before using the formula.
- 5
A cone with the same base radius and vertical height as a cylinder has one-third of that cylinder's volume: (1/3) pi r^2 h.
- 6
A sphere of radius r has volume (4/3) pi r^3, so radius errors grow cubically in the answer.
- 7
One litre equals 1000 cm^3 and one cubic metre equals 1000 litres, which is central in tank and capacity questions.
- 8
In recasting questions, equate total volume; surface area normally changes and should not be equated.
Continue studying
Volume, capacity and unit discipline
Volume is the amount of three-dimensional space occupied by a solid. Capacity is the amount a hollow object can hold. In CET volume questions, both ideas often use the same numerical formula, but the wording decides whether the answer is a solid volume or usable inner capacity. A cube of side 5 cm has volume 125 cm^3; a cubical container with inner side 5 cm has capacity 125 cm^3, which is 0.125 litre.
The main discipline is dimensional. Length is measured in cm, m, or km; area is measured in square units; volume is measured in cubic units. If all dimensions are in cm, the volume is in cm^3. If all dimensions are in m, the volume is in m^3. Mixed dimensions must be converted before substitution. For capacity, 1 litre = 1000 cm^3, 1 millilitre = 1 cm^3, and 1 m^3 = 1000 litres. A tank measuring 2 m by 1 m by 50 cm should be treated as 2 m by 1 m by 0.5 m, giving 1 m^3 or 1000 litres.
Exam use: first bring all dimensions into one unit, then apply the volume formula, and only then convert capacity if the answer is asked in litres.
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