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The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al space it occupies, often quantified numerically. One-dimensional figures (such as lines
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
) and two-dimensional shapes (such as square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s) are assigned zero volume in the three-dimensional space. Volume is commonly presented in units such as mL or cm3 (milliliters or cubic centimeters).

Volumes of some simple shapes, such as regular, straight-edged and circular shapes can be easily calculated using arithmetic formula
Formula

In mathematics and in the sciences, a formula is a concise way of expressing information symbolically , or a general relationship between quantities....
s.






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The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al space it occupies, often quantified numerically. One-dimensional figures (such as lines
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
) and two-dimensional shapes (such as square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s) are assigned zero volume in the three-dimensional space. Volume is commonly presented in units such as mL or cm3 (milliliters or cubic centimeters).

Volumes of some simple shapes, such as regular, straight-edged and circular shapes can be easily calculated using arithmetic formula
Formula

In mathematics and in the sciences, a formula is a concise way of expressing information symbolically , or a general relationship between quantities....
s. More complicated shapes can be calculated by integral calculus if a formula exists for its boundary. The volume of any shape can be determined by displacement
Displacement (fluid)

In fluid mechanics, displacement occurs when an object is immersed in a fluid, pushing it out of the way and taking its place. The volume of the fluid displaced can then be measured, as in the illustration, and from this the volume of the immersed object can be deduced ....
.

In differential geometry, volume is expressed by means of the volume form
Volume form

In mathematics, a volume element provides a means for integration a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates....
, and is an important global Riemannian
Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, manifold with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smooth function from point to point....
 invariant
Invariant

Invariant and invariance may have several meanings, among which are:* Invariant , an expression whose value doesn't change during execution ...
.

Volume is a fundamental parameter in thermodynamics
Thermodynamics

In physics, thermodynamics is the study of the conversion of heat energy into different forms of energy ; different energy conversions into heat energy; and its relation to macroscopic variables such as temperature, pressure, and volume....
 and it is conjugate
Conjugate variables (thermodynamics)

In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume....
 to pressure
Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
.

Related terms


The density
Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
 of an object is defined as mass per unit volume. The inverse of density is specific volume
Specific volume

Specific volume is the volume occupied by a unit of mass of a material. It is equal to the inverse of density. Specific volume may be expressed in , or ....
 which is defined as volume divided by mass.

Volume and capacity
Capacity

Capacity is the ability to hold, receive or absorb, or a measure thereof, similar to the concept of volume.Capacity may also refer to:*Capacity utilization, the point of production at which a firm or industry's average costs begin to rise, usually because some factor is fixed ....
 are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in liter
Litér

Lit?r is a village in Veszpr?m , Hungary.External links ...
s or its derived units), and volume being how much space an object displaces (commonly measured in cubic meters or its derived units).

Volume and capacity are also distinguished in a capacity management
Capacity management

Capacity Management is a process used to manage information technology . Its primary goal is to ensure that IT capacity meets current and future business requirements in a cost-effective manner....
 setting, where capacity is defined as volume over a specified time period.

Traditional cooking measures

measure US Imperial metric
teaspoon
Teaspoon

Kari is extraterrestial.A teaspoon, a type of cutlery , is a small spoon, commonly silver and part of a place setting, suitable for stirring and sipping the contents of a cup of tea or coffee....
 		 1/6 U.S. fluid ounce (about 4.929 mL) 1/6 Imperial fluid ounce (about 4.736 mL) 5 mL
tablespoon
Tablespoon

A tablespoon is a type of large spoon usually used for serving. A tablespoonful, an amount equal to the capacity of one tablespoon, is commonly used as a unit of measurement of volume used in Cooking weights and measures ....
 = 3 teaspoons 		˝ U.S. fluid ounce (about 14.79 mL) ˝ Imperial fluid ounce (about 14.21 mL) 15 mL
cup
Cup (unit)

The cup is a Units of measurement for volume, used in cooking to measure bulk foods, such as granulated sugar , and liquids . It is in common use in the United States and nations influenced by them, such as Japan....
  		8 U.S. fluid ounces or ˝ U.S. liquid pint (about 237 mL) 8 Imperial fluid ounces or 2/5 fluid pint (about 227 mL) 250 mL
In the UK, a tablespoon can also be five fluidrams (about 17.76 mL).

Volume formulas

Shape Equation Variables
A cube a = length of any side (or edge)
A rectangular prism: l = length, w = width, h = height
A cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
:		 r = radius of circular face, h = height
A general prism
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
:		 B = area of the base, h = height
A sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
:		 r = radius of sphere
which is the integral
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 of the Surface Area
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
 of a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
An ellipsoid
Ellipsoid

An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
:		 a, b, c = semi-axes of ellipsoid
A pyramid
Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
:		 A = area of the base, h = height of pyramid
A cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
 (circular-based pyramid):		 r = radius of circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 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.

For their volume formulas, see the articles on tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 and parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
.


Volume formula derivation


Sphere


The volume of a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 is the integral
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 of infinitesimal circular slabs of thickness . The calculation for the volume of a sphere with center 0 and radius is as follows.

The radius of the circular slabs is

The surface area of the circular slab is .

The volume of the sphere can be calculated as
Now
and


Combining yields

This formula can be derived more quickly using the formula for the sphere's surface area
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
, which is . The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to

=

Cone

The volume of a cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
 is the integral
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 of infinitesimal circular slabs of thickness . The calculation for the volume of a cone of height , whose base is centered at (0,0) with radius is as follows.

The radius of each circular slab is , and varying linearly in between -- that is,

The surface area of the circular slab is then

The volume of the cone can then be calculated as

Substituting gives an integral with reversed limits, and : that is,

Swapping the limits makes this

Integrating gives us

See also

  • Area
    Area

    Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
  • Conversion of units
    Conversion of units

    Conversion of units refers to conversion factors between different units of measurement for the same quantity....
  • Density
    Density

    The density of a material is defined as its mass per unit volume. The symbol of density is ....
  • Dimensional weight
    Dimensional weight

    Dimensional weight, used in shipping and Cargo, is a billing technique which takes into account the volume of a package....
  • Dimensioning
    Dimensioning

    Dimensioning is the process of measuring the cubic space that a package or object occupies. It is the method of calculating dimensional weight for the storage, handling, transporting and invoicing of goods....
  • Length
    Length

    Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end....
  • Mass
    Mass

    In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
  • Orders of magnitude (volume)
    Orders of magnitude (volume)

    The pages linked in the right-hand column contain lists of volumes that are of the same orders of magnitude . Rows in the table represent increasing powers of a thousand....
  • Volume form
    Volume form

    In mathematics, a volume element provides a means for integration a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates....
  • Weight
    Weight

    In the physical sciences, weight is a measurement of the gravitational force acting on an object. Near the surface of the Earth, the Earth's gravity is approximately constant; this means that an object's weight is roughly proportional to its mass....


External links