Volume is the quantification of the three-dimensional space a substance occupies. The SI unit for volume is the cubic meter, or m3. By convention, the volume of a container is typically its capacity, and how much fluid it is able to hold, rather than the amount of space that the actual container displaces. Volumes of many shapes can be calculated by using well-defined formulas. In some cases, more complicated shapes can be broken down into simpler aggregate shapes, and the sum of their volumes is used to determine total volume. The volumes of other even more complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary. Beyond this, shapes that cannot be described by known equations can be estimated using mathematical methods, such as the finite element method. Alternatively, if the density of a substance is known, and is uniform, the volume can be calculated using its weight. This calculator computes volumes for some of the most common simple shapes.

Sphere

A sphere is the three-dimensional counterpart of a two-dimensional circle. It is a perfectly round geometrical object that, mathematically, is the set of points that are equidistant from a given point at its center, where the distance between the center and any point on the sphere is the radius r. Likely the most commonly known spherical object is a perfectly round ball. Within mathematics, there is a distinction between a ball and a sphere, where a ball comprises the space bounded by a sphere. Regardless of this distinction, a ball and a sphere share the same radius, center, and diameter, and the calculation of their volumes is the same. As with a circle, the longest line segment that connects two points of a sphere through its center is called the diameter, d. The equation for calculating the volume of a sphere is provided below:

volume = 

4

3

πr3

EX: Claire wants to fill a perfectly spherical water balloon with radius 0.15 ft with vinegar to use in the water balloon fight against her arch-nemesis Hilda this coming weekend. The volume of vinegar necessary can be calculated using the equation provided below:

volume = 4/3 × π × 0.153 = 0.141 ft3

Cone

A cone is a three-dimensional shape that tapers smoothly from its typically circular base to a common point called the apex (or vertex). Mathematically, a cone is formed similarly to a circle, by a set of line segments connected to a common center point, except that the center point is not included in the plane that contains the circle (or some other base). Only the case of a finite right circular cone is considered on this page. Cones comprised of half-lines, non-circular bases, etc. that extend infinitely will not be addressed. The equation for calculating the volume of a cone is as follows:

volume = 

1

3

πr2h

where r is the radius and h is the height of the cone

EX: Bea is determined to walk out of the ice cream store with her hard-earned $5 well spent. While she has a preference for regular sugar cones, the waffle cones are indisputably larger. She determines that she has a 15% preference for regular sugar cones over waffle cones and needs to determine whether the potential volume of the waffle cone is ≥ 15% more than that of the sugar cone. The volume of the waffle cone with a circular base with radius 1.5 in and height 5 in can be computed using the equation below:

volume = 1/3 × π × 1.52 × 5 = 11.781 in3

Bea also calculates the volume of the sugar cone and finds that the difference is < 15%, and decides to purchase a sugar cone. Now all she has to do is use her angelic, childlike appeal to manipulate the staff into emptying the containers of ice cream into her cone.

Cube

A cube is the three-dimensional analog of a square, and is an object bounded by six square faces, three of which meet at each of its vertices, and all of which are perpendicular to their respective adjacent faces. The cube is a special case of many classifications of shapes in geometry, including being a square parallelepiped, an equilateral cuboid, and a right rhombohedron. Below is the equation for calculating the volume of a cube:

volume = a3
where a is the edge length of the cube

EX: Bob, who was born in Wyoming (and has never left the state), recently visited his ancestral homeland of Nebraska. Overwhelmed by the magnificence of Nebraska and the environment unlike any other he had previously experienced, Bob knew that he had to bring some of Nebraska home with him. Bob has a cubic suitcase with edge lengths of 2 feet, and calculates the volume of soil that he can carry home with him as follows:

volume = 23 = 8 ft3

 

Conversions are performed by using a conversion factor. By knowing the conversion factor, converting between units can become a simple multiplication problem:

S * C = E

Where S is our starting value, C is our conversion factor, and E is our end converted result.

To simply convert from any unit into cubic meters, for example, from 10 liters, just multiply by the conversion value in the right column in the table below.

10 L * 0.001 [ (m3) / (L) ] = 0.01 m3

To convert from m3 into units in the left column divide by the value in the right column or, multiply by the reciprocal, 1/x.

0.01 m3 / 0.001 [ (m3) / (L) ] = 10 L

To convert among any units in the left column, say from A to B, you can multiply by the factor for A to convert A into m/s2 then divide by the factor for B to convert out of m3.  Or, you can find the single factor you need by dividing the A factor by the B factor.

For example, to convert from liters to gallons you would multiply by 0.001 then divide by 0.003785412.  Or, multiply by 0.001/0.003785412 = 0.26417203. So, to convert directly from L to gal you multiply by 0.26417203.