Skip to main content
CalcMax

Sphere Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

523.5988 cm³

Volume

Total surface area
314.1593 cm²
Radius
5.0000 cm
Diameter
10.0000 cm
Circumference
31.4159 cm

A sphere calculator takes any one of three measurements of a ball — its radius, its diameter or its circumference — and returns five: the volume, the surface area, and all three of those lengths. The shape of the form is worth noticing. Only one box should be filled in, and the page works from whichever it is, because the three lengths are all the same measurement of the same ball wearing different clothes: measure around the equator and you have as good a description of the ball as measuring from the centre to the skin. What is not symmetric is the other direction. A circle's four measurements are peers, and the circle page next door will start from any of them. A sphere's are not: the volume and the surface area are things you work out, never things you supply, because going backwards from a volume means a cube root and going backwards from an area means a square root, and both of those questions already have pages of their own. So the page has three boxes and five readings, and the two extra readings are the point of it. The volume is how much the ball holds or displaces — how much air is in a football, how much water a spherical tank takes, how much material a ball bearing is made of. The surface area is how much skin it has — how much paint, how much leather, how much plating. Those two are what makes a sphere a three-dimensional object, and they are what the flat shapes on the pages either side of this one have nothing to compare to.

Common balls, entered from each of the three measurements

Given asRadius (cm)Diameter (cm)Circumference (cm)Surface area (cm²)Volume (cm³)
r51031.4159314.1593523.5988
d61237.6991452.3893904.7787
C3.18316.366220127.324135.0949
r102062.83191256.63714188.7902
r126.283212.56644.1888
r0.513.14163.14160.5236
r00000
d50100314.159331415.9265523598.7756

Eight balls, and the first column says which of the three boxes was filled in — without it the table would not read, since all five numbers are printed for every row and there is nothing else to say which measurement was the input. The first three rows are entered three different ways on purpose: a 5 cm radius, a 12 cm diameter and a 20 cm circumference, so that the three entry points are all visible side by side. The first row is the ball the page loads with, and its circumference of 31.4159 is exactly what the circle page prints for a radius of 5 — a circumference does not care whether it is drawn flat or around a ball. The third row is the one to read if you plan to do this arithmetic 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 — a difference in the last digit, and the page is right, because the true answer belongs to the 20 you typed rather than to the rounded radius on display. The fifth row is the unit ball, where the surface area is 4π and the volume is 4π/3. The sixth row is the coincidence worth stopping on: at a radius of one half the surface area and the circumference are both 3.1416, because 4πr² and 2πr cross exactly there. The seventh row is a radius of zero, where every column is zero and the answer is real rather than missing. The last row is a diameter of 100, ten times the radius of the first row, and the scaling is visible in it: ten times the radius gives ten times the circumference, a hundred times the surface area and a thousand times the volume. 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 A = 4πr² d = 2r C = 2πr

Radius
The distance from the centre to the surface, in centimetres. The one number a sphere is really built on — everything else on this page, including the other two inputs, is a way of expressing it
Diameter
The distance straight through the middle, in centimetres, which is twice the radius. The measurement to use when you have the ball in your hands and a rule across it, which is how most balls are actually described
Circumference
The distance all the way around the widest part, in centimetres, which is 2πr. The measurement to use when you have a tape and can only get it around the outside — a tree trunk, a pipe, or a ball too large to reach across
Volume
How much space the ball takes up, in cubic centimetres. It is the only reading here that goes with the cube of the radius, which is why it runs away from the others so quickly as balls get bigger
Surface area
How much skin the ball has, in square centimetres, which is four times the area of a circle of the same radius. It is exactly the same as the area of the curved surface of the cylinder that would just contain the ball
4⁄3 πr³
The volume formula, and the four thirds is the part worth remembering. A sphere holds two thirds of the cylinder that just fits around it — the result Archimedes asked to have carved on his tombstone
4πr²
The surface area, and the plain 4 is worth noticing next to the volume's four thirds. It is the area of four flat circles of the same radius, which is a coincidence of the calculus rather than anything you could see by unfolding it
Four decimal places
How wide every reading is written. Only the radius and the diameter can be exact; the circumference, the surface area and the volume all carry a π and the volume carries a cube as well, so their last digits are a display width rather than a claim

This page is for anything round in three dimensions, and the two readings it adds over the circle page are the ones that matter for real objects. Volume is for capacity and displacement: how much air a football holds, how much a spherical tank or a gas holder takes, how much a ball bearing weighs once you know its material, how much buoyancy a float provides. Surface area is for anything that covers the ball: paint, leather panels on a football, plating on a bearing, the cooling surface of a droplet, or the dose of a drug carried on the surface of a particle. There is also a use that runs the other way, and it is the commonest one of all: you have a ball and a tape measure, and what you want is a number you cannot reach. Wrap the tape around the equator, put the circumference in, and the page gives you the diameter and the volume without your having to cut the ball open — which is exactly how the size of a tree, a pipe or a planet is arrived at.

Worked examples

  1. A ball of radius 5

    1. Diameter: 2 × 5 = 10
    2. Circumference: 2 × π × 5 = 10π = 31.4159…
    3. Surface area: 4 × π × 5² = 100π = 314.1593…
    4. Volume: (4 ÷ 3) × π × 5³ = 500π ÷ 3 = 523.5987…, which rounds to 523.5988

    The input the page loads with. The two π readings are worth reading as whole numbers of π: the surface area is 100π and the volume is 500π/3. It is also the row where the sphere's numbers can be checked against the circle page's — this ball's circumference of 31.4159 is exactly what the circle page prints for a radius of 5, because a circumference does not care whether it is drawn on a flat page or around a ball. The volume is where the third dimension shows up.

  2. A ball of diameter 12

    1. Radius: 12 ÷ 2 = 6
    2. Circumference: 2 × π × 6 = 12π = 37.6991…
    3. Surface area: 4 × π × 6² = 144π = 452.3893…
    4. Volume: (4 ÷ 3) × π × 6³ = 288π = 904.7787…

    The entry to use when the ball is in front of you: a diameter is what a rule across the middle gives, and it is how footballs, bearings and planets are actually specified. Everything here is a whole multiple of π — 12π, 144π, 288π — which is what happens whenever the radius comes out a whole number, and is a good sign the arithmetic is right.

  3. A ball measured around its equator

    1. Radius: 20 ÷ (2π) = 3.1830988…, which rounds to 3.1831
    2. Diameter: 2 × 3.1830988… = 6.3661977…, which rounds to 6.3662
    3. Surface area: 4 × π × 3.1830988…² = 400 ÷ π = 127.3239…, which rounds to 127.324
    4. Volume: (4 ÷ 3) × π × 3.1830988…³ = 4000 ÷ 3π² = 135.0948…, which rounds to 135.0949

    The row to read if you ever plan to do this arithmetic yourself, because it is the one where rounding early bites. The radius printed above is 3.1831, which is already rounded — cube that and you get a volume of 135.0950 rather than the 135.0949 the page reports. Both numbers are defensible and only one is right: the true answer belongs to the 20 you typed in, and the 3.1831 is a display of it rather than the thing itself. It is also the entry to use for anything you can only get a tape around, such as a tree trunk or a pipe.

  4. A ball of radius one half

    1. Diameter: 2 × 0.5 = 1
    2. Circumference: 2 × π × 0.5 = π = 3.1416…
    3. Surface area: 4 × π × 0.5² = π = 3.1416…
    4. Volume: (4 ÷ 3) × π × 0.5³ = π ÷ 6 = 0.5236…

    The row worth stopping on: the circumference and the surface area come out as the same number, 3.1416, and it is not a slip. The two formulas are 2πr and 4πr², and at a radius of one half they cross — 2π × ½ and 4π × ¼ are both π. Below this radius the surface area is the smaller of the two; above it, the larger. It is the sort of coincidence that only turns up when a length and an area happen to be measured in the same units, and on this page they are.

  5. A ball of radius zero

    1. Diameter: 2 × 0 = 0
    2. Circumference: 2 × π × 0 = 0
    3. Surface area: 4 × π × 0² = 0
    4. Volume: (4 ÷ 3) × π × 0³ = 0

    A ball of no size is a single point, and all five readings are zero. Zero is a legal input rather than an empty box, so a row of zeros is a real answer — the page distinguishes it from leaving all three boxes blank, which shows nothing 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 the Earth itself is not one, and neither is a hemisphere, which is a sphere cut in half with an extra flat face whose area is not included here. The volume and the surface area cannot be entered — they are answers, not inputs — so a ball specified by how much it holds has to be worked back to a radius elsewhere first. All five readings are in centimetres, square centimetres and 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. Four decimal places is a display width rather than a claim about precision: only the radius and the diameter can be exact, and the other three carry a π, with the volume carrying a cube as well. The readings are rounded independently from the exact radius, so a value printed on this page will not always reproduce another one printed beside it — the true answer belongs to the number you entered. Exactly one box may be filled; two at once is rejected rather than silently resolved, because a radius of 5 and a diameter of 10 are the same ball but a radius of 5 and a diameter of 12 are not, and the page would rather say so than pick. Nothing here handles the thickness of a hollow shell, the material a ball is made of, or its weight.

Frequently asked questions

Why can I only fill in one box?
Because the three boxes describe the same ball. The radius, the diameter and the circumference are one measurement wearing three different outfits — fill any one and the page knows the ball. Filling two is rejected rather than resolved, because a radius of 5 and a diameter of 10 agree while a radius of 5 and a diameter of 12 do not, and there is no honest way for the page to guess which of the two you meant.
Why can't I enter the volume and get the radius?
Because that is a different question, and it needs a cube root rather than a multiplication. The same goes for entering a surface area, which needs a square root. Both are perfectly reasonable things to want, but they are separate problems, and this page takes the three lengths as its inputs so that it can give the volume and the area as answers. Working backwards is a page of its own.
The surface area and the circumference came out as the same number. Is that wrong?
No — at a radius of one half they genuinely are equal. The circumference is 2πr and the surface area is 4πr², and at r = ½ those are π and π. Below that radius the surface area is smaller than the circumference and above it the surface area is larger. They are different kinds of quantity, so the equality is a numerical coincidence rather than a statement that a length equals an area.
Where does the four thirds in the volume formula come from?
It makes the sphere exactly two thirds of the cylinder that just fits around it. Archimedes worked this out and thought enough of it to ask for the sphere and the cylinder to be carved on his tombstone. It is also why the volume formula looks so much heavier than the ones for boxes and prisms: a curved surface cannot be filled by stacking flat slices without the calculus.
Which measurement should I use for a ball I can only reach around?
The circumference. Wrap a tape around the widest part, put that number in the circumference box, and the page gives you the diameter and the radius along with the volume and the surface area. It is the standard way to size a tree trunk, a pipe, or anything too large or too awkward to measure across, and it is the reason the circumference is an input here rather than just another reading.
Should the answer be in cubic centimetres or litres?
The page prints cubic centimetres, and the conversion is the usual one: a thousand cubic centimetres is a litre, so a volume of 523.5988 cm³ is about 0.5236 litres. For smaller volumes, a millilitre is the same as a cubic centimetre. The unit dropdown changes what you type in, not what comes out, so if you enter a diameter in inches the answer will still be in centimetres.

References

Related calculators

Sphere Volume CalculatorHow much space a sphere takes, from its radius or diameter.Circle CalculatorRadius, diameter, circumference and area from any one of them.Surface Area CalculatorSurface area of spheres, cones, cylinders and prisms.Cylinder Volume CalculatorVolume of a cylinder, plus the areas of its ends and its curved side.