Sphere Volume Calculator
Result
Volume
A sphere volume calculator takes one measurement of a ball — its radius, its diameter or its circumference — and returns how much space the ball takes up, in cubic centimetres. It is the sphere page next door with everything else stripped away. That page takes the same three inputs and answers five questions at once; this one answers the one that people actually arrive with, which is how much a ball holds, displaces or is made of. The formula is four thirds times π times the radius cubed, and the shape of it is worth a moment. Almost every volume formula in this group is a base multiplied by a height, because almost every solid in this group can be sliced into flat layers that stack up. A sphere cannot be. Its layers change size at every height, the slices near the middle are wide and the ones near the poles are tiny, and adding them up is what the calculus was invented for. The four thirds is what that sum comes to, and the satisfying way to remember it is that it makes the ball exactly two thirds of the smallest cylinder that would fit around it. Archimedes worked this out and asked for the sphere and the cylinder to be carved on his tombstone. The other thing worth knowing is how fast this number grows. Volume goes with the cube of the radius, so doubling a ball's size multiplies its volume by eight and tripling it multiplies it by twenty-seven. That is why a small change in a diameter makes a large change in what a tank holds, and why the answer here can be very large very quickly.
The space a ball takes up, entered from each of the three measurements
| Given as | Radius (cm) | Diameter (cm) | Volume (cm³) |
|---|---|---|---|
| r | 5 | 10 | 523.5988 |
| d | 6 | 12 | 904.7787 |
| C | 3.1831 | 6.3662 | 135.0949 |
| r | 10 | 20 | 4188.7902 |
| r | 1 | 2 | 4.1888 |
| r | 0.5 | 1 | 0.5236 |
| r | 0 | 0 | 0 |
| d | 50 | 100 | 523598.7756 |
Eight balls and four columns — two columns narrower than the sphere page next door, which prints the same three inputs and then gives the surface area and the circumference as well. The first column says which box was filled in, and without it the table would not read, since the radius and the diameter are both printed on every row. The first row is the ball the page loads with, and the fourth is one of the rows worth comparing against it: a radius of 5 against a radius of 10, where the radius has doubled and the volume has gone from 523.5988 to 4188.7902 — eight times, which is the cube law in one step. The third row is the one to read if you plan to do this by hand: entered as a circumference of 20 the volume is 135.0949, while cubing the rounded radius of 3.1831 printed beside it gives 135.0950. The fifth row is the unit ball, where the volume is 4π/3 and the sphere page's surface area for the same ball is 4π — a third of it, numerically. The seventh row is a radius of zero, where the ball holds nothing and the answer is real rather than missing. The last row is a diameter of 100, where the volume runs to more than half a million cubic centimetres — half a cubic metre of ball, and a reminder of how fast this quantity leaves the range of everyday units. Every number here is recomputed from its entry when the page is built, in centimetres, and the four decimals are the same four decimals the results panel uses.
Formula
V = 4⁄3 × πr³ r = d ÷ 2 r = C ÷ 2π
- Radius
- The distance from the centre to the surface, in centimetres. The one number the volume is actually made of — the other two boxes are only other ways of arriving at it
- Diameter
- The distance straight through the middle, in centimetres, which the page halves to get the radius. The box to use when the ball is in front of you and a rule across it is the easiest measurement to take
- Circumference
- The distance all the way around the widest part, in centimetres, which the page divides by 2π to get the radius. The box to use when a tape around the outside is all you can manage
- Volume
- How much space the ball occupies, in cubic centimetres. The only output on the page, and the only quantity here that grows with the cube of the size rather than with the size itself
- 4⁄3 πr³
- The formula, and the four thirds is the part that means something. It makes the ball two thirds of the cylinder that just fits around it, which is a fact about spheres rather than a quirk of the arithmetic
- r³
- The radius cubed, and the reason this page can produce big numbers from modest inputs. Doubling the radius multiplies the volume by eight; the same is true whether the ball is a marble or a planet
- Four decimal places
- How wide the reading is written. The volume always carries a π and a cube, so it is never exact unless the radius is zero — the four decimals are a display width rather than a claim about precision
This page is for capacity, displacement and material, which between them cover most of what anyone wants from a ball. Capacity: how much a spherical tank, gas holder or water tower holds, how much air is in a football, how much ice cream a scoop delivers. Displacement: how much water a float or a hull pushes aside, which is the same as its volume and is what determines whether it floats and how much it can carry. Material: how much steel is in a ball bearing, how much concrete in a dome's worth of fill, how much a ball of dough or clay weighs once you know its density. It is also the shape of a good many things that are only approximately spherical, and the approximation is usually good enough — an egg, a droplet, a hailstone, a planet that bulges slightly at the equator. And there is the reverse use: given a volume that a container has to hold, this is the page that tells you how big the sphere would have to be, since a cube root is all that stands between the two.
Worked examples
A ball of radius 5
- Cube the radius: 5³ = 125
- Multiply by π and by four thirds: (4 ÷ 3) × π × 125 = 500π ÷ 3
- 500π ÷ 3 = 523.5987…, which rounds to 523.5988
The input the page loads with, and the row to check the formula against: the volume is 500π/3, so the whole thing is one clean multiple of π. Five centimetres is a bit smaller than a tennis ball, and the answer — a little over half a litre — is a useful thing to have a feel for, since it is the sort of quantity a recipe or a small container deals in.
A ball of diameter 12
- Radius: 12 ÷ 2 = 6
- Cube the radius: 6³ = 216
- Multiply by π and by four thirds: (4 ÷ 3) × π × 216 = 288π
- 288π = 904.7786…, which rounds to 904.7787
The entry to reach for when the ball is in your hands, since a diameter is what a rule across the middle gives and it is how most balls are specified. The answer is 288π, a whole multiple of π again, which happens whenever the radius comes out a whole number. Compare it with the row above: the radius grew by a fifth, from 5 to 6, and the volume grew by more than two thirds, from 523.5988 to 904.7787.
A ball measured around its equator
- Radius: 20 ÷ (2π) = 3.1830988…
- Cube the radius: 3.1830988…³ = 32.2519…
- Multiply by π and by four thirds: (4 ÷ 3) × π × 32.2519… = 4000 ÷ 3π²
- 4000 ÷ 3π² = 135.0948…, which rounds to 135.0949
The row to read if you plan to do this arithmetic yourself, because it is where rounding early goes wrong. The sphere page prints the radius for this ball as 3.1831, which looks precise and is not: cube that and multiply it out and you get 135.0950. The page reports 135.0949 and the page is right, because the true answer belongs to the 20 you entered, not to the rounded radius on display. The two differ in the last digit and nowhere else, which is exactly the sort of error that survives a sanity check.
A ball of radius one
- Cube the radius: 1³ = 1
- Multiply by π and by four thirds: (4 ÷ 3) × π × 1 = 4π ÷ 3
- 4π ÷ 3 = 4.18879…, which rounds to 4.1888
The unit ball, and the row that shows what the four thirds actually is: the volume is 4π/3, and the surface area the sphere page prints for the same ball is 4π. The volume is a third of the surface area numerically, which is a coincidence of the radius being one and not a general rule — but it is a convenient pair of numbers to remember, and it makes the two formulas easy to tell apart.
A ball of radius zero
- Cube the radius: 0³ = 0
- Multiply by π and by four thirds: (4 ÷ 3) × π × 0 = 0
A ball of no size is a point and holds nothing. Zero is a legal input rather than an empty box, so this is a real answer — the page distinguishes it from leaving all three boxes blank, which shows no result at all because there is nothing to work from.
Limitations
The page is for spheres only: every point on the surface the same distance from the centre. An egg, a rugby ball or a slightly flattened planet is not one, and neither is a hemisphere — which is a sphere cut in half and holds exactly half of what is computed here, though its surface area is not half. The volume cannot be entered, so a ball specified by how much it holds has to be worked back to a radius elsewhere first. The reading is in cubic centimetres whatever unit the dropdown is set to, so a diameter entered in inches comes back as a centimetre answer you have to convert — and since this is a volume, the conversion is a cube: cubic inches to cubic centimetres is a factor of about 16.4, not 2.54. Four decimal places is a display width rather than a claim about precision, since the volume always carries both a π and a cube. Exactly one box may be filled; two at once is rejected rather than silently resolved. Nothing here handles the wall thickness of a hollow shell, the material a ball is made of, or how much it weighs, and the page gives no surface area — the sphere page next door prints that alongside the volume for the same three inputs.
Frequently asked questions
- How is this different from the sphere calculator?
- The inputs are identical — the same three boxes, the same rule of filling in exactly one. The difference is what comes back. The sphere page prints five readings: the volume, the surface area and the three lengths. This page prints the volume and nothing else, because that is the one people usually arrive wanting, and stripping the rest away makes it the only thing on screen.
- Why does the volume grow so much faster than the diameter?
- Because it goes with the cube of the radius while the diameter goes with the radius itself. Double a ball's diameter and its volume goes up eight times; treble it and the volume goes up twenty-seven times. That is why a small change in the size of a tank makes a large change in what it holds, and why the answer here can be very large from a modest-looking input.
- Where does the four thirds come from?
- It makes the ball exactly two thirds of the smallest cylinder that would fit around it — same radius, and a height of twice the radius. Archimedes proved this and asked for the sphere and the cylinder to be carved on his tombstone. It is also why the formula carries a fraction at all: a sphere is not a stack of flat layers, and the calculus that adds up its curved ones is what produces the four thirds.
- My own arithmetic gives 135.0950 but the page says 135.0949. Which is right?
- The page. The radius for that ball is 10/π, which is 3.1830988…, and the radius printed on screen is rounded to 3.1831. Cubing the rounded figure is what produces the 135.0950 — the difference is in the last digit and it comes entirely from rounding early. The true answer belongs to the number you typed in, and the page keeps the exact radius internally until the very end.
- Can I get the radius back from a volume?
- Not on this page — the volume is an output, not an input. It is perfectly possible, since the radius is the cube root of three times the volume over four π, but it is a different operation and the page does not do it. If you know how much a container has to hold and want to know how big a ball that would be, that is the calculation to do separately.
- Should the answer be in litres?
- The page prints cubic centimetres, and a thousand of them make a litre — so 523.5988 cm³ is about 0.524 litres, and one millilitre is one cubic centimetre. For larger spheres the numbers get big quickly: a ball a metre across holds over half a cubic metre, which is more than five hundred litres. The unit dropdown changes what you type in rather than what comes out.
References
- Sphere — the ball itself, with the volume formula this page computes and the two-thirds relation to the cylinder that just contains it — Wolfram MathWorld (United States)
- Volume — what a volume is in general and how it is defined for a curved solid, which is the part of the sphere that cannot be reached by stacking flat slices — Wolfram MathWorld (United States)
- Cylinder — the solid a sphere fits inside, whose volume the sphere takes exactly two thirds of, and the shape most nearly comparable to it — Wolfram MathWorld (United States)
- 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); the volume of a sphere is part of the compulsory-education mathematics curriculum, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部