A cylinder, radius 3 and height 5
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r = 3, h = 5V = πr²h = π · 3² · 5 = π · 9 · 5 = 45π, which is about 141.37. Square the radius before multiplying by the height; doing it the other way round gives 225π, five times too big.
Volume Calculator
Six solids, each with its formula written out before it is used.
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Every volume formula is an area times a height
A prism - a box, a cylinder, anything with the same cross-section all the way up - has a volume of base area times height. That is one idea covering two of the six solids here: a box is length × width × height because its base area is length × width, and a cylinder is πr²h because its base is a circle of area πr². Learning the pattern beats learning the formulas separately.
The ones that taper - the cone and the pyramid - take exactly a THIRD of the prism that would contain them. Same base, same height, one third of the volume. That factor of 1/3 is not arbitrary and it is the single most forgotten thing on this topic; a cone the same size as a cylinder holds a third as much, which is easier to remember once you have poured one into the other.
The sphere is the odd one out at 4/3πr³, and it is worth keeping exact for the same reason as everything else here: π is irrational, so 36π is an answer and 113.1 is a rounding of it. This page keeps the symbol, reports the decimal beside it, and gives you the surface area and the slant height wherever those are exact too.
What to notice
- For anything prism-shaped, volume is base area × height. Recognising that turns four formulas into one idea plus the right area formula.
- A cone or pyramid is one third of the prism with the same base and height. Forgetting the 1/3 is the classic error and makes the answer exactly three times too big.
- Every measurement must be in the same unit before you start, and the volume is then in that unit CUBED. Mixing centimetres and metres is a factor-of-a-million mistake.
- Doubling one dimension doubles the volume; doubling all three multiplies it by eight. That is why volume grows so much faster than length.
- Pi belongs in the answer, not in the working. 45π is exact, and multiplying it out to 141.37 at the end costs nothing and is easy to check.
- A cone's slant height is not its height. The slant is the hypotenuse of the radius and the height, which is why it comes out as a square root - and it is what the curved surface area needs, not the volume.
How to find a volume
Identify the solid and write down its formula
Do this before touching a number. Cube s³, box lwh, cylinder πr²h, sphere 4/3πr³, cone 1/3πr²h, square pyramid 1/3s²h.
Check that the units match
Convert everything to the same unit first. This is the step that silently wrecks answers, because the arithmetic afterwards looks perfectly fine.
Substitute the measurements
Write the formula out with your numbers in it before working anything out. On a marked question, that substituted line is usually worth a mark of its own.
Work out the powers before multiplying
For a cylinder of radius 3 and height 5: square the 3 first to get 9, then 9 · 5 = 45, giving 45π. Multiplying before squaring gives a different and wrong number.
Give the answer in cubic units
A radius in centimetres gives a volume in cubic centimetres. Leave π in place unless a decimal was asked for.
The six formulas, with a worked value
Each formula and what it gives for a small example. Notice the 1/3 on the two solids that taper.
| Solid | Formula | Example | Exact volume | Decimal |
|---|---|---|---|---|
| Cube | s³ | s = 4 | 64 | 64 |
| Box | l · w · h | 2, 3, 4 | 24 | 24 |
| Cylinder | πr²h | r = 3, h = 5 | 45π | 141.3717 |
| Sphere | 4/3 πr³ | r = 3 | 36π | 113.0973 |
| Cone | 1/3 πr²h | r = 3, h = 4 | 12π | 37.6991 |
| Square pyramid | 1/3 s²h | s = 6, h = 4 | 48 | 48 |
| Sphere | 4/3 πr³ | r = 2 | 32π/3 | 33.5103 |
| Cylinder | πr²h | r = 1/2, h = 4 | π | 3.1416 |
Worked examples
A sphere of radius 3
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r = 3V = 4/3 πr³ = 4/3 · π · 27 = 36π, about 113.1. The 27 and the 4/3 cancel neatly here, which is why radius 3 is the example every textbook uses. Its surface area is 4πr² = 36π too - a coincidence unique to r = 3.
A cone, radius 3 and height 4
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r = 3, h = 4V = 1/3 πr²h = 1/3 · π · 9 · 4 = 12π, about 37.7. Compare it with the cylinder of the same base and height, which holds 36π: the cone is exactly a third. Its slant height is √(3² + 4²) = 5, which the volume does not need but the curved surface does.
A cube of side 4
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s = 4V = s³ = 4³ = 64, and no π anywhere. Its surface area is 6s² = 96, because a cube has six identical square faces. Both answers are whole numbers, which makes this the useful one to check your method against.
When the answer keeps a fraction: a sphere of radius 2
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r = 2V = 4/3 · π · 8 = 32π/3, which is about 33.51. The 3 does not divide 32, so the exact answer keeps the fraction - writing 33.5 loses it, and in any question that continues with this value the error travels.
Common mistakes
- Forgetting the 1/3 on a cone or pyramid. The answer comes out exactly three times too big, which is the easiest error to spot once you know to look.
- Mixing units. Convert first: a radius in centimetres with a height in metres produces a number that means nothing at all.
- Multiplying before squaring. In πr²h it is the radius that is squared, not the product - π · (3 · 5)² is a completely different quantity.
- Using the diameter as the radius. It doubles r, which multiplies a cylinder's volume by four and a sphere's by eight.
- Confusing volume with surface area. One is space inside and comes out in cubic units; the other is skin and comes out in square units.
- Using the slant height in a cone's volume. The volume needs the perpendicular height; the slant is for the curved surface area.
Volume FAQ
What is the formula for the volume of a cylinder?
V = πr²h - the area of the circular base times the height. For a radius of 3 and a height of 5 that is π · 9 · 5 = 45π, or about 141.37. Square the radius before multiplying by the height.
What is the volume of a sphere?
V = 4/3 πr³. A sphere of radius 3 holds 4/3 · π · 27 = 36π, about 113.1. It is the one formula here that cannot be read as base area times height, so it has to be memorised.
Why is there a 1/3 in the cone and pyramid formulas?
Because a cone or pyramid holds exactly a third of the prism with the same base and the same height. Pour a full cone into a cylinder of matching size and it fills one third of it - the factor is a fact about tapering, not a fudge.
What units does a volume come out in?
Whatever unit your measurements were in, cubed. Centimetres give cubic centimetres, metres give cubic metres. All the measurements must be converted to the same unit BEFORE you calculate, or the answer is meaningless.
How do I find the volume of a rectangular box?
Multiply the three dimensions: V = l · w · h. It is base area times height like any prism, since the base area of a box is l · w. A 2 by 3 by 4 box holds 24 cubic units.
What is the difference between volume and surface area?
Volume is the space inside and is measured in cubic units; surface area is the total area of the outside and is measured in square units. This page reports both wherever the surface area is exact - for a cube, a box, a cylinder and a sphere.
What is the slant height of a cone?
The distance from the tip down the sloping side to the rim: √(r² + h²), by the Pythagorean theorem on the radius and the height. It is longer than the height, it is what the curved surface area formula needs, and it plays no part in the volume.
Why keep pi symbolic instead of multiplying it out?
Because 45π is exact and 141.37 is a rounding. Exam questions frequently ask for an answer in terms of π precisely because it can be checked without a calculator, and rounding early in a multi-step problem loses precision the later steps need.
