Volume Calculator

Eleven calculators sit above this, one for each of the shapes that come up most often. Fill in the measurements, pick the unit beside each one, and the volume comes back with the working shown. Units can be mixed freely, so a radius in inches and a height in feet gives the right answer.

What volume measures

Volume is the amount of three-dimensional space something occupies. The SI unit is the cubic meter, m³, though litres and gallons are more usual for liquids and cubic feet for building materials.

For a container there is a distinction worth keeping straight. Its capacity is how much it holds, measured from the inside. Its displacement is how much space the object itself takes up, measured from the outside, walls included. A steel drum has a capacity of about 200 litres and displaces slightly more. Use inside measurements for what fits in, outside measurements for what it occupies.

Complicated objects can usually be broken into simple ones and their volumes added, which is how the capsule below is built out of a cylinder and two half-spheres. Where no formula exists, integral calculus handles anything with a describable boundary, and numerical methods such as finite element analysis handle the rest. If the density is known and uniform, weighing the object and dividing by density gives the volume without measuring it at all.

Sphere

A sphere is the set of all points a fixed distance from a centre, that distance being the radius r. Mathematically a sphere is the surface and a ball is the solid inside it, though the volume calculation is the same either way. The longest straight line through it, the diameter, is 2r.

$$\text{volume}=\frac{4}{3}\pi r^3$$

A basketball has a radius of about 0.12 m, so it holds 4/3 × π × 0.12³ = 0.00724 m³, which is 7.24 litres of air. Note how hard volume responds to the radius: double the radius and the volume goes up eight times, because r is cubed. That is why a pizza of twice the diameter is four times the food and a ball of twice the radius is eight times the water.

Cone

A cone tapers from a circular base to a single point, the apex. Only the right circular cone is handled here, meaning the apex sits directly above the centre of the base. The height h is measured straight up from the base plane to the apex, not along the sloping side.

$$\text{volume}=\frac{1}{3}\pi r^{2}h$$

A waffle cone with a 1.5 inch radius and a height of 5 inches holds 1/3 × π × 1.5² × 5 = 11.781 in³. The one-third is not arbitrary: a cone holds exactly a third of the cylinder that would enclose it, which you can check by filling one with water and pouring it into the other three times.

Cube

A cube is bounded by six equal square faces meeting at right angles. It is the three-dimensional counterpart of a square, and a special case of several broader families, including the square parallelepiped and the equilateral cuboid.

$$\text{volume}=a^3$$

A packing crate with 2 ft edges holds 2³ = 8 ft³. Because all three dimensions are the same, a small measuring error is magnified: getting the edge wrong by 5% puts the volume out by roughly 16%.

Cylinder

A cylinder here means a right circular cylinder, two parallel circular ends joined by a surface at right angles to them. It is the shape of tins, tanks, pipes and drums, which makes it the most-used formula on this page.

$$\text{volume}=\pi r^{2}h$$

A barrel with a 3 ft radius and a height of 4 ft holds π × 3² × 4 = 113.097 ft³. If you have the diameter rather than the radius, halve it first. Using diameter in place of radius is the single most common mistake with this formula and it inflates the answer four times over.

Rectangular tank

A rectangular tank is a cube with its three dimensions free to differ, bounded by six rectangular faces at right angles. Multiply the three together in any order.

$$\text{volume}=\text{length}\times \text{width}\times \text{height}$$

An aquarium 4 ft long, 2 ft wide and 2 ft deep holds 16 ft³, which is about 119.7 US gallons. Fish tanks are quoted by capacity, so measure the inside, and remember that water sits below the rim: a tank filled to 90% of its depth holds 90% of that figure.

Capsule

A capsule is a cylinder with a hemisphere on each end. Since two hemispheres make one sphere, the volume is a cylinder plus a whole sphere of the same radius.

$$\text{volume}=\pi r^{2}h+\frac{4}{3}\pi r^3$$

With a radius of 1.5 ft and a cylindrical section 3 ft long, that is 4/3 × π × 1.5³ + π × 1.5² × 3 = 35.343 ft³. Read the h input carefully: it is the straight section only, so the overall end-to-end length is h + 2r. Propane tanks and pressure vessels are capsules for a reason, since curved ends carry pressure far better than flat ones.

Spherical cap

A spherical cap is the piece of a ball cut off by a single flat plane. Cut through the centre and the cap is a hemisphere. Three measurements describe it and any two fix the third: the base radius r, the ball radius R, and the cap height h.

$$\text{volume}=\frac{1}{3}\pi h^2(3R-h)$$

Converting between them:

$$h=R\pm \sqrt{R^2-r^2}$$$$R=\frac{h^2+r^2}{2h}$$$$r=\sqrt{2Rh-h^2}$$

The plus-or-minus in the first is real rather than a technicality. A given base radius and ball radius describe two different caps: a shallow dish and the large remainder of the ball. With r = 2 and R = 3 the two heights are 0.76393202250021 and 5.2360679774998, giving volumes of 5.0333597369843 and 108.06397579225. The calculator above reports both, since only you know which one you meant.

A domed roof 2 m high on a sphere of radius 6 m has a base radius of √(2 × 6 × 2 − 2²) = 4.472 m and encloses 1/3 × π × 2² × (18 − 2) = 67.021 m³.

Conical frustum

A frustum is what remains of a cone once the top has been sliced off parallel to the base. Buckets, lampshades, plant pots and most disposable cups are frustums, which makes this more useful in daily life than the full cone.

$$\text{volume}=\frac{1}{3}\pi h(r^2+rR+R^2)$$

A bucket 0.35 m deep with a 0.25 m radius at the bottom and 0.2 m at the top holds 1/3 × π × 0.35 × (0.2² + 0.2 × 0.25 + 0.25²) = 0.05589 m³, or 55.9 litres. The r × R term in the middle is what people forget; leaving it out understates the answer badly.

Ellipsoid

An ellipsoid is a sphere stretched by different amounts along three perpendicular axes. When all three differ it is called tri-axial; when two match it is a spheroid, the shape of a rugby ball or a squashed planet. The a, b and c in the formula are semi-axes, measured from the centre outwards, so they are half the full widths.

$$\text{volume}=\frac{4}{3}\pi abc$$

Semi-axes of 1.5, 2 and 5 inches give 4/3 × π × 1.5 × 2 × 5 = 62.832 in³. Set all three equal and the formula collapses back to the sphere, which is a useful check that you have entered semi-axes and not diameters.

Square pyramid

A pyramid connects a polygonal base to an apex. A square pyramid has a square base, and a right pyramid has its apex directly above the centre of that base. The height is the perpendicular distance from the base plane to the apex, and the volume formula holds wherever the apex sits horizontally, which is Cavalieri's principle at work.

$$\text{volume}=\frac{1}{3}{a}^{2}h$$

A mud pyramid with 5 ft base edges and a height of 12 ft contains 1/3 × 5² × 12 = 100 ft³. As with the cone, the one-third means a pyramid holds a third of the box that would enclose it.

Tube

A tube, or pipe, is a hollow cylinder. Its volume is the outer cylinder minus the inner one, and because pipes are specified by diameter rather than radius the formula is written that way, dividing by 4 to account for it.

$$\text{volume}=\pi \frac{{d_1}^2-{d_2}^2}{4}l$$

A pipe with an outer diameter of 3 ft, an inner diameter of 2.5 ft and a length of 10 ft contains π × (3² − 2.5²) ÷ 4 × 10 = 21.598 ft³ of material. That figure is the wall, which is what you want when ordering concrete or working out weight. For how much the pipe carries, run the cylinder calculator on the inner diameter instead.

Common volume units

Unit	cubic meters	milliliters
milliliter (cubic centimeter)	0.000001	1
cubic inch	0.00001639	16.39
pint	0.000473	473
quart	0.000946	946
liter	0.001	1,000
gallon	0.003785	3,785
cubic foot	0.028317	28,317
cubic yard	0.764555	764,555
cubic meter	1	1,000,000
cubic kilometer	1,000,000,000	1015

Two traps worth knowing. The US gallon and the imperial gallon are different sizes, 3.785 litres against 4.546, so a figure in gallons means little without the country. And cubed units scale by the cube of the length conversion: there are 3 feet in a yard but 27 cubic feet in a cubic yard, and 100 centimetres in a metre but a million cubic centimetres in a cubic metre.

Reading your result

Check the units on your inputs first. The calculators above take a unit per measurement and report the answer in the cube of whatever the first field uses, so mixing inches and feet is fine as long as each dropdown matches the number beside it.

Then check radius against diameter. Most of the round shapes here take a radius, while the tube deliberately takes diameters because that is how pipe is sold. Feeding a diameter into a radius field overstates the volume by a factor of four for anything with r² in it and eight for a sphere.

For a container, decide whether you want capacity or displacement before measuring, and measure inside or outside to match. If your object is not on this list, try splitting it into pieces that are: a silo is a cylinder plus a spherical cap, and a pencil is a cylinder plus a cone.