Volume Calculator
Below you’ll find calculators for determining the volume of a variety of frequently‑used shapes. Enter the required measurements and press the “Calculate” button.
Sphere Volume Calculator
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Cone Volume Calculator
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Cube Volume Calculator
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Cylinder Volume Calculator
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Rectangular Tank Volume Calculator
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Capsule Volume Calculator
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Spherical Cap Volume Calculator
Please provide any two values below to calculate.
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Conical Frustum Volume Calculator
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Ellipsoid Volume Calculator
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Square Pyramid Volume Calculator
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Tube Volume Calculator
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Volume measures how much three‑dimensional space an object occupies. In the "SI" system it is expressed in cubic metres (m3). Usually we talk about a container’s volume as the amount of fluid it can hold, not the space the container itself displaces. Many solid figures have straightforward formulas that give their volume. For more complex bodies we can split them into simpler parts, sum the individual volumes, or apply integral calculus when a boundary equation is known. When analytical formulas are unavailable, numerical techniques such as the finite‑element method can provide estimates. If the material density is uniform and known, the volume can also be obtained from its mass. The tools below handle the most typical basic shapes.
Sphere
A sphere is the three‑dimensional analogue of a circle – an ideal round solid where every point on its surface lies at the same distance, the radius r, from a central point. In everyday language the term “ball” often refers to a solid that fills the interior of a sphere, but mathematically both share the same centre, radius and diameter d. The longest straight line through the centre, connecting two opposite points on the surface, is the diameter. The formula used to compute a sphere’s volume is shown below.
| volume = |
|
πr3 |
Example: Claire wishes to prepare a spherical water balloon with a radius of 0.15 ft, filling it with vinegar for her upcoming water‑balloon showdown with Hilda. The needed amount of vinegar can be determined with the formula given next.
volume = 4/3 × π × 0.153 = 0.141 ft3
Cone
A cone is a solid that narrows smoothly from a typically circular base up to a single tip called the apex (or vertex). Geometrically it can be imagined as a set of line segments that all meet at a common apex while their other ends trace a circle in a plane; the apex itself does not lie in that plane. This page only deals with finite right circular cones – infinite cones, cones with non‑circular bases, or other variations are omitted. The volume of such a cone is calculated with the following equation.
| volume = |
|
πr2h |
where r is the radius and h is the height of the cone
Example: Bea plans to leave the ice‑cream shop having spent her $5 wisely. Although she prefers classic sugar cones, waffle cones are noticeably larger. She assumes a 15 % preference for sugar cones over waffle ones and wants to know whether the waffle cone’s volume is at least 15 % greater. Using a waffle cone with a circular base of radius 1.5 in and a height of 5 in, the volume can be computed with the formula displayed below.
volume = 1/3 × π × 1.52 × 5 = 11.781 in3
Bea then determines the volume of the sugar cone, finds the difference to be less than 15 %, and decides to stick with the sugar cone. She now just needs to charm the staff into filling it completely with ice cream.
Cube
A cube is the three‑dimensional counterpart of a square: a solid bounded by six equal square faces, three meeting at each corner, with each face orthogonal to its neighbours. It can be thought of as a special case of several geometric families, such as a square parallelepiped, an equilateral cuboid, or a right rhombohedron. The formula for a cube’s volume is provided below.
volume = a3
where a is the edge length of the cube
Example: Bob, a lifelong Wyoming resident, recently traveled to his family’s roots in Nebraska. Struck by the state’s vast landscapes, he decides to bring a piece of it home. He owns a cubic suitcase with each side measuring 2 ft and wants to know how much soil he can pack inside. The volume is calculated as follows:
volume = 23 = 8 ft3
Cylinder
A cylinder, in its most basic sense, consists of all points that remain at a fixed distance from a given straight line (the axis). In everyday usage the term usually refers to a right circular cylinder, where the two ends are circles aligned with an axis that is perpendicular to their planes. The cylinder is defined by its height h and radius r. Its volume can be obtained with the equation shown below.
volume = πr2h
where r is the radius and h is the height of the tank
Example: Caelum plans to construct a sandcastle inside his living room. As a strong proponent of reuse, he salvaged three cylindrical drums discarded at an unlawful dump, rinsed away the toxic residues with soap and water. Each drum measures 3 ft in radius and 4 ft tall, and Caelum will calculate the sand capacity of each using the formula shown.
volume = π × 32 × 4 = 113.097 ft3
He successfully builds a sandcastle in his house, and as an added bonus, manages to save electricity on nighttime lighting, since his sandcastle glows bright green in the dark.
Rectangular Tank
A rectangular tank is essentially a stretched cube, with each edge allowed to differ in length. It consists of six orthogonal faces, three intersecting at every corner. Below is the formula used to compute its volume.
volume= length × width × height
Example: Darby is a cake enthusiast who works out four hours daily to balance her indulgence. She intends to trek the Kalalau Trail on Kauai, but doubts her stamina without sufficient cake. Opting to travel light, she aims to fill a rectangular backpack measuring 4 ft × 3 ft × 2 ft with cake. The precise amount of cake that fits is computed with the following formula.
volume = 2 × 3 × 4 = 24 ft3
Capsule
A capsule consists of a right circular cylinder capped with two hemispheres (each half of a sphere). Therefore, its total volume is obtained by adding the volume of the cylinder to the volumes of the two hemispherical ends.
| volume = πr2h + |
|
πr3 = πr2( |
|
r + h) |
where r is the radius and h is the height of the cylindrical portion
EX: Given a capsule with a radius of 1.5 ft and a height of 3 ft, determine the volume of melted milk chocolate m&m's that Joe can carry in the time capsule he wants to bury for future generations on his journey of self-discovery through the Himalayas:
volume = π × 1.52 × 3 + 4/3 ×π ×1.53 = 35.343 ft3
Spherical Cap
A spherical cap represents the slice of a sphere cut off by a single plane. When the cutting plane goes through the sphere’s center, the cap becomes a hemisphere. A related shape is a spherical segment, formed by two parallel planes intersecting the sphere at distinct radii. The cap’s volume formula is a special case of the segment formula with the second radius set to zero. See the illustration in the calculator for reference.
| volume = |
|
πh2(3R - h) |
When any two of the three parameters are supplied, the tool determines the missing dimension and the corresponding volume. The conversion relationships between height and radii are displayed below.
Given r and R: h = R ± √R2 - r2
| Given r and h: R = |
|
where r is the radius of the base, R is the radius of the sphere, and h is the height of the spherical cap
Example: Jack is determined to outplay his buddy James on the golf course to wow Jill. Instead of training, he trims a flawless spherical cap from the top of James’s ball and wants to know how much material is required to fill the gap and alter the ball’s mass. With the ball’s radius at 1.68 in and the removed cap’s height at 0.3 in, the volume is computed below.
volume = 1/3 × π × 0.32 (3 × 1.68 - 0.3) = 0.447 in3
Unfortunately for Jack, James happened to receive a new shipment of balls the day before their game, and all of Jack's efforts were in vain.
Conical Frustum
A conical frustum is what stays of a cone after it is sliced by two parallel planes. This tool evaluates the volume of a right circular frustum. Common examples of frustums are lampshades, buckets, and certain glassware. Its volume is given by the formula below.
| volume = |
|
πh(r2 + rR + R2) |
where r and R are the radii of the bases, h is the height of the frustum
Example: Bea manages to scoop ice cream into a sugar cone and consumes it until the remaining ice cream sits level with the cone’s rim. Just as she prepares to finish, her brother bites off the bottom of the cone, creating a perfectly parallel cut. She is left with a right conical frustum of ice cream and must determine how much ice cream she needs to eat quickly. Using a frustum height of 4 in and radii of 1.5 in and 0.2 in, the volume is calculated as follows:
volume=1/3 × π × 4(0.22 + 0.2 × 1.5 + 1.52) = 10.849 in3
Ellipsoid
An ellipsoid is the three‑dimensional analogue of an ellipse, formed by stretching a sphere along three perpendicular axes. Its center is where these axes intersect, and the axes themselves are called the principal axes. When all three axes differ, the shape is a tri‑axial ellipsoid. The volume can be computed with the formula below.
| volume = |
|
πabc |
where a, b, and c are the lengths of the axes
Example: Xabat is a meat lover, but his mother says he eats too much and will only let him fill an ellipsoid‑shaped bun with meat. To maximize the amount, Xabat hollows out the bun. The bun’s three semi‑axes measure 1.5 in, 2 in and 5 in, so the meat volume he can pack is computed as follows:
volume = 4/3 × π × 1.5 × 2 × 5 = 62.832 in3
Square Pyramid
A pyramid is a solid created by joining a polygonal base to a single apex point. When the base is a square we speak of a square pyramid. If the apex lies directly above the centroid of the base, the pyramid is called right. Regardless of the apex’s position, as long as the height is taken as the perpendicular distance from the base plane to the apex, the volume can be expressed as:
Generalized pyramid volume:
| volume = |
|
bh |
Square pyramid volume:
| volume = |
|
a2h |
Example: Wan, who loves ancient Egyptian architecture, gathers his three siblings—Too, Tree, and Fore—to act as laborers. Together they construct a mud pyramid whose base edges measure 5 ft and whose height is 12 ft. The volume can be found with the square‑pyramid formula:
volume = 1/3 × 52 × 12 = 100 ft3
Tube Pyramid
A tube, also known as a pipe, is essentially a hollow cylinder used to convey fluids or gases. Its volume is calculated similarly to a solid cylinder (V = π r² h), but we use the tube’s outer and inner diameters together with its length. First compute the volume of the outer cylinder, then subtract the inner cylinder’s volume. The resulting expression for a tube’s volume is shown below:
| volume = π |
|
l |
where d1 is the outer diameter, d2 is the inner diameter, and l is the length of the tube
Example: Beulah, an eco‑focused contractor, has been asked to install a conduit across a creek so a road can remain accessible without hindering the fish. She chooses a pipe with an outer diameter of 3 ft, an inner diameter of 2.5 ft, and a length of 10 ft. The amount of low‑impact concrete required is calculated as follows:
| volume = π × |
|
× l0 = 21.6 ft3 |
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 |