Volume Calculator

Volume counts unit cubes, the way area counts unit squares. A box l by w by h holds l times w cubes in each layer and h layers, so its volume is l × w × h. That is less a formula than the definition, and every other formula in this calculator is a way of doing the same count for a shape that does not stack into neat layers.

The first family is prisms: any solid with the same cross-section all the way along. A box, a cube, a cylinder and a triangular prism are all prisms, and all have the same formula: the area of the cross-section times the length. For a cylinder the cross-section is a circle, πr², so the volume is πr²h. For a triangular prism it is half base times height, times the length. Once you see that, you have every prism, including ones with cross-sections no table lists. A pipe is a hollow cylinder: what it holds uses the inside diameter, and its wall is the outer cylinder minus the inner one.

The second family is solids that taper to a point: cones and pyramids. Each is exactly one third of the prism with the same base and height, which is where the 1/3 in both formulas comes from. Three identical square pyramids fit together into a cube, which is the hands-on proof; the general one needs calculus, or Cavalieri's principle, and it holds for any base shape. A frustum (a cone with its top sliced off, the shape of a bucket or a lampshade) is the large cone minus the small one, which tidies into (1/3)πh(R² + Rr + r²). The tempting shortcut of averaging the top and bottom areas and multiplying by the height gives too much, and the error grows the more the shape tapers.

The sphere is Archimedes' result, and the one he asked to have on his tomb. Fit a sphere inside a cylinder that just contains it, radius r and height 2r. The sphere is exactly two-thirds of the cylinder's volume, (2/3) × πr² × 2r = (4/3)πr³, and its surface area is exactly the cylinder's curved side, 2πr × 2r = 4πr². A hemisphere is half the volume, but its total surface is not half the sphere's: cutting it exposes a flat face, so it is 3πr². A capsule is a cylinder with a hemisphere at each end, and the two ends make one sphere.

An ellipsoid is a sphere stretched by a different amount along each axis. Stretching multiplies volume by the stretch factor, so (4/3)πr³ becomes (4/3)πabc, where a, b and c are the semi-axes (half the full length, width and height). Its surface area is a different matter: there is no formula for it in elementary functions at all. The exact value needs elliptic integrals, which this calculator evaluates numerically rather than using one of the published approximations.

When the measurements are whole numbers, the answer is shown as an exact multiple of π: a sphere of radius 6 has a volume of exactly 288π. That form is the one to keep if more arithmetic follows, because 904.78... is already rounded and 288π is not. It is also the form a textbook answer key uses.

Units matter more in three dimensions than in two. A cubic conversion cubes the length factor: a yard is 3 feet, so a cubic yard is 27 cubic feet, not 9 and not 3. A liter is a cube 10 cm on a side, 1,000 cubic centimeters, and a cubic meter is 1,000 liters. The US gallon is defined as exactly 231 cubic inches.

The same cube law explains the table under the result. Double every length and the volume goes up eight times while the surface area goes up only four. As things get bigger, surface shrinks relative to volume. That is why a large pot of water takes so long to cool and why crushed ice melts faster than a block of the same weight.