Volume Calculator
Volume and surface area for nine solids, from the dimensions you can measure on the outside. Every answer comes with the formula it used and your own numbers substituted into it — and for the cone and the sphere, with the cylinder they sit inside.
Cylinder
Cylinder, cone, and sphere are the interesting three — the same radius and height put them in a fixed 3 : 1 : 2 ratio.
Centre to edge across the circular end.
The length of the straight side.
In cubic units of whatever you measured in — cubic inches if you typed inches, cubic metres if you typed metres.
- Volume
- 282.7433388
- Surface area
- 245.044227
- Volume formula, with your numbers
- V = πr²h = π × 3² × 10 = 282.7433388
- Surface area formula, with your numbers
- S = 2πr(r + h) = 2 × π × 3 × (3 + 10) = 245.044227
- Every length here is a plain number with no unit attached. The volume comes back in the cube of whatever you measured in, so the geometry is the same whether you work in millimetres or miles.
- A cone with this same radius and height holds exactly one third of this cylinder, and a sphere of this radius fills exactly two thirds of the cylinder that just contains it. Switch the solid above and watch the ratio land on the nose.
Nine solids, nine formulas, no units
Each solid has its own closed-form volume, and all of them are exact. The lengths you type carry no unit, so the answer comes back in the cube of whatever you measured in — cubic millimetres if you worked in millimetres, cubic feet if you worked in feet. The one rule is that every length in a single calculation has to be in the same unit.
Vbox = lwh • Vcyl= πr²h • Vcone= ⅓πr²h • Vsphere = ⁴⁄₃πr³ • Vpyr= ⅓a²h • Vprism= ½bhL • Vtorus = 2π²Rr²- l, w, h
- The three edges of a rectangular box
- r
- Radius — centre to surface, which is half the distance all the way across
- h (cone, pyramid)
- Vertical height from the centre of the base to the tip, not the slanted side
- a
- One edge of a square pyramid’s base
- b, h, L (prism)
- Base and perpendicular height of the triangular end face, and how far it is extruded
- R, r (torus)
- Ring radius from the centre of the hole to the centre of the tube, and the tube’s own radius
Surface areas come back alongside, with one honest exception: the triangular prism. Its base and height fix the area of the triangular face without fixing its outline, so the lateral surface cannot be worked out from what you entered, and the tool says so rather than guessing at an isosceles triangle.
3 : 1 : 2 — the cylinder, the cone, and the sphere
Take a cylinder of radius 5 and height 10. Its volume is π × 5² × 10 = 785.3981634.
Now put a cone inside it, standing on the same circular base and reaching the same height. Its volume is 261.7993878 — exactly a third.
Now put a sphere of radius 5 inside instead. It touches the cylinder all the way round the middle and touches both ends, because a cylinder of radius 5 and height 10 is precisely the smallest one that will hold it. Its volume is 523.5987756 — exactly two thirds.
And 261.7993878 + 523.5987756 = 785.3981634. The cone and the sphere, together, fill the cylinder exactly. One third plus two thirds, with nothing left over. Change the radius, change the height, change the units: the three numbers move and the ratio 3 : 1 : 2 does not.
The surfaces do the same thing, which is the half that almost never gets mentioned. The sphere’s surface is 4π × 5² = 314.1592654. The curved side of the cylinder — ignore the two flat ends for a second — is 2π × 5 × 10, which is also 314.1592654. They are not close. They are the same number, and they are the same number for every radius, because the cylinder’s height is 2r and 2πr × 2r is 4πr². Peel the label off a tin and it has exactly the area of the ball that fits inside. That identity is why the Lambert cylindrical projection is an equal-area world map: you can wrap a sphere in a cylinder and project outwards without changing any area at all.
Add the cylinder’s two flat ends, 2π × 5² = 157.0796327, and its total surface is 471.238898. The sphere’s 314.1592654 divided by that is, once again, exactly two thirds.
Archimedes proved the sphere result in On the Sphere and the Cylinder around 225 BC — no calculus, no algebraic notation, working by exhaustion arguments that squeeze the answer between two sequences until nothing else is possible. He thought it was the best thing he had done and asked for a sphere inscribed in a cylinder to be carved on his tomb. Cicero, posted to Sicily as quaestor in 75 BC, wrote that he hunted down the grave outside Syracuse, found it overgrown with thorns, and recognised it by that diagram.
The practical payoff: you only have to remember πr²h. A cone with the same base and height is a third of it, a sphere is two thirds of the cylinder that just contains it, and a square pyramid is a third of its box for exactly the same reason a cone is a third of its cylinder — a cube cuts cleanly into three identical pyramids. Four formulas collapse into one and three fractions.
A cylinder of radius 3 and height 10, checked by hand
The default here is a cylinder with a radius of 3 and a height of 10. The base is a circle of area π × 3² = 28.27433388, and the volume is that area carried up 10 units: π × 9 × 10 = 282.7433388. Note that the radius is half the way across, not all the way — measuring a 6-unit-wide tube and typing 6 is the most common mistake on this page, and it quadruples the answer.
The surface area is the curved side plus the two circular ends: 2πrh = 2 × π × 3 × 10 = 188.4955592 for the side, and 2πr² = 2 × π × 9 = 56.54866776 for the ends, giving 245.044227 in total. Both terms appear in the single expression 2πr(r + h), which is what the calculator shows.
Switch the solid to a cone and keep the same radius and height. The volume drops to 94.24777961, and 94.24777961 × 3 = 282.7433388 — the cylinder, back again. The result panel shows that enclosing cylinder and the ratio explicitly, so the check is not something you have to do on paper.
One more, for the shape people most often get wrong. A capsule with a radius of 3 and a straight section 10 long is not a cylinder with something vague added: it is exactly a cylinder plus one complete sphere, because its two domed ends are hemispheres that fit together. So the volume is 282.7433388 + 113.0973355 = 395.8406744, and shortening the straight section to nothing would leave the sphere alone, correctly.
Which volume tool you actually want
If you have the dimensions of a solid and want the space it encloses, you are on the right page; if you already have a volume and need it in different units, you want the converter; and if you are buying concrete, you want the one that counts bags.
- Volume converter — for units rather than geometry. Litres to gallons, millilitres to fluid ounces, cubic feet to cubic metres, and the gap between US and imperial measures. Use it once you have a volume and need it expressed differently.
- Concrete calculator — for materials rather than mathematics. Same geometry underneath, but it converts to cubic yards for the truck, counts 40, 60 and 80 lb bags, adds a waste allowance you can argue with, and flags orders under the usual one-yard delivery minimum.
The three overlap in a search box and nowhere else. Keeping them apart is what lets this page carry nine solids and the Archimedes relationship instead of a bag table and a units grid crammed in beside them.
What this calculator assumes
- Every length is a plain number with no unit, and all the lengths in one calculation are in the same unit. Volume comes back in that unit cubed and surface area in that unit squared. Nothing here converts between units.
- Solids are right, not oblique. A cylinder’s axis is perpendicular to its base and a cone’s apex sits directly over the centre of its base. A leaning cone has the same volume but a different surface area, and it is not modelled here.
- Heights are vertical. The height of a cone or a pyramid is measured straight up from the centre of the base to the tip, not along the slanted side. Using the slant instead overstates both.
- Radii are half-widths. Cylinders, cones, spheres and capsules are described by the distance from the centre to the surface, not the distance all the way across.
- A capsule’s height is its straight middle section only. The two domed ends add one full sphere on top of it.
- A torus is a ring torus: the tube radius never exceeds the ring radius. Beyond that the surface passes through itself, the hole closes, and 2π²Rr² counts the overlap twice — so the calculator refuses rather than returning a plausible wrong number.
- Solids are solid. Nothing here models wall thickness, hollow shells, or how efficiently spheres pack into a box.
- Results are rounded to ten significant figures rather than a fixed number of decimal places, because cubing the input range spans from billionths to quintillions and no single decimal place suits both ends.
Volume calculator FAQ
Why is a cone exactly one third of the cylinder around it?
It is easiest to believe from the flat-sided version. A cube can be cut into exactly three identical square pyramids: put the apex at one corner of the cube and each pyramid takes one of the three faces that do not touch that corner as its base. Three pieces, one cube, so each pyramid is a third of a box with the same base and the same height. Now let the base become a circle and nothing about the argument changes — every horizontal slice of a cone is a scaled copy of its base in exactly the way every slice of a pyramid is, so the same third applies. A cylinder of radius 5 and height 10 holds 785.398; the cone inside it holds 261.799, which is that figure divided by three.
What was carved on Archimedes' tomb?
A sphere inside a cylinder, with the ratio 2:3 alongside it. Archimedes proved in On the Sphere and the Cylinder, around 225 BC, that a sphere occupies exactly two thirds of the volume of the smallest cylinder that will contain it — and that its surface area is exactly two thirds of that cylinder's total surface as well. He regarded it as his finest result and asked for the diagram on his grave. Cicero, serving as quaestor in Sicily in 75 BC, wrote that he found the tomb outside Syracuse choked with brambles and identified it by that carving. Nearly two thousand years before calculus, and it is still the cleanest fact in solid geometry.
Which volume tool do I actually want?
If you have the dimensions of a solid and want the space it encloses, this one. If you already have a volume and need it in different units — litres to gallons, millilitres to fluid ounces — use the volume converter. If you are working out how much concrete to order for a slab, a footing, or a column, use the concrete calculator: that one converts to cubic yards, counts 40, 60 and 80 lb bags, adds a waste allowance, and flags the short-load minimum, none of which is geometry. The three overlap in a search box and nowhere else.
What units does this use?
Whatever you used. Every length here is a plain number with no unit attached, and the volume comes back in the cube of the unit you measured in: enter centimetres and the answer is cubic centimetres, enter feet and it is cubic feet. Surface area comes back in that unit squared. The only rule is that all the lengths in one calculation must be in the same unit — convert first if they are not.
Why does the triangular prism show no surface area?
Because you did not give it enough information, and no amount of arithmetic can recover the rest. A base and a perpendicular height fix the area of the triangular end face and say nothing about its outline: slide the apex sideways along a line parallel to the base and the area holds steady while the two other sides stretch as far as you like. The volume is therefore exact — cross-sectional area times length — and the lateral surface genuinely is not determined. Rather than quietly assume the triangle is isosceles and print a number that is probably wrong, the tool says so.
How do I find the volume of something that is not one of these shapes?
Two practical routes. If it can be cut into shapes that are on this list, do that and add the volumes — a cylinder plus a cone is a silo, a box plus a triangular prism is a house. If it cannot, use displacement: fill a container to a mark, lower the object in, and measure how much the level rises. That is the method Archimedes is supposed to have leapt out of the bath over, and for an awkward casting or a hand-shaped part it is still faster and more accurate than any decomposition.
Why does a capsule's volume not depend on where you split it?
Because the two domed ends of a capsule are hemispheres of the same radius, and two hemispheres are one sphere however long the straight section between them is. So the volume is always the cylindrical middle plus one complete sphere: πr²h + (4 ÷ 3)πr³. The same tidiness holds for the surface, 2πr(2r + h) — the lateral surface of the cylinder plus the surface of one whole sphere. Making the straight section shorter shrinks the first term to nothing and leaves you with a plain sphere, which is the right answer.
Sources and review notes
- MacTutor History of Mathematics Archive, University of St Andrews — Archimedes of Syracuse, On the Sphere and the Cylinder, and Cicero’s account of finding the tomb
- Wolfram MathWorld — Torus, for 2π²Rr² and the ring-torus condition that the tube radius must not exceed the ring radius
- NIST Special Publication 811, Guide for the Use of the International System of Units — the conventions behind treating a volume as a length cubed
The geometry on this page is exact and has no data behind it to go stale — a sphere has been two thirds of its cylinder since 225 BC and will stay that way. Every figure here is an identity rather than a measurement, so there is nothing to re-check as prices, standards, or conventions move.