Volumes of straight-edged and circular shapes are calculated using arithmetic formulae. Volumes of other curved shapes are calculated using integral calculus, by approximating the given body with a large amount of small cubes or concentric cylindrical shells, and adding the individual volumes of those shapes. The volume of irregularly shaped objects can be determined by displacement. If an irregularly shaped object is less dense than the fluid, you will need a weight to attach to the floating object. A sufficient weight will cause the object to sink. The final volume of the unknown object can be found by subtracting the volume of the attached heavy object and the total fluid volume displaced.
Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in liters or its derived units), and volume being how much space an object displaces (commonly measured in cubic metrics or its derived units). The volume of a dispersed gas is the capacity of its container. If more gas is added to a closed container, the container either expands (as in a balloon) or the pressure inside the container increases.
Volume and capacity are also distinguished in a capacity management setting, where capacity is defined as volume over a specified time period.
|Common equations for volume:|
|A cube:||s = length of any side|
|A rectangular prism:||l = length, w = width, h = height|
|A cylinder (circular prism):||r = radius of circular face, h = height|
|Any prism that has a constant cross sectional area along the height**:||A = area of the base, h = height|
|A sphere:||r = radius of sphere|
which is the integral of the Surface Area of a sphere
|An ellipsoid:||a, b, c = semi-axes of ellipsoid|
|A pyramid:||A = area of the base, h = height of pyramid|
|A cone (circular-based pyramid):||r = radius of circle at base, h = distance from base to tip|
|Any figure (calculus required)||h = any dimension of the figure, A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. This will work for any figure if its cross-sectional area can be determined from h (no matter if the prism is slanted or the cross-sections change shape). ^*|
(The units of volume depend on the units of length - if the lengths are in meters, the volume will be in cubic meters, etc)
Traditional cooking measures for volume also include:
The density of an object is defined as mass per unit volume.
|Shape||Volume formula derivation|
|Sphere||The volume of a sphere is the integral of infinitesimal circular slabs of width . The calculation for the volume of a sphere with center 0 and radius r is as follows.
The radius of the circular slabs is
The surface of the circular slab is
The volume of the sphere can be calculated as
Transform the variable of integration from by so that transforms to . Since should take the value +1 when takes the value , the integral boundaries become -1 and +1, we get . (This substitution is difficult to
Thus, the sphere volume amounts to Vsphere = =
This formula can be derived more quickly using the formula for the sphere surface area, which is . The volume of the sphere consists of layers of infinitesimal spherical slabs, and THE sphere volume is equal to