Volume

In mathematics or daily life, the volume, of an object, is a quantification of how much space it occupies. In terms of volume measurements, volume may also be termed capacity.

The volume of a solid object is a numerical value given to describe the three-dimensional concept of how much space it occupies. One-dimensional objects (such as lines) and two-dimensional objects (such as squares) are assigned zero volume in three-dimensional space.

The volume in acoustics refers to loudness. It is a common term for the amplitude of sound.

Less commonly, in mathematics, volume can refer to the amount of space an n-dimensional object fills up, for some n > 3. Volumes are defined by means of integral calculus, by the decomposition of complex sets into small volume elements. Volume (Cx³) is the antiderivative of area (Cx²). More simply, for a perfect closed curve, which is the sphere in three dimensions, the volume is the simple integral of the aurface area. Thus, the surface area of a sphere is 4πr², and the volume is 4/3πr³.

Table of contents

1 Volume formulae 2 Volume measures: SI 3 Volume measures: USA 4 Volume measures: UK 5 Volume measures: cooking 6 Relationship to density 7 Volume comparisons 8 External link

Volume formulae

Common equations for volume:

• A cube: s³ (where s is the length of a side)
• A rectangular prism: l w h (length, width, height)
• A cylinder: π r² h (r = radius of circular face, h = distance between faces)
• A sphere: 4 π r³ / 3 (r = radius of sphere)
• A cone: π r² h / 3 (r = radius of circle at base, h = distance from base to tip)
• any prism that has a constant cross sectional area along the height**: A h (A = area of the base, h = height)
• any figure (calculus required): ∫ A dh (where h is any dimension of the figure, and A is the area of the cross sections perpendicular to h described as a function of the position along h) (this will work for any figure (no matter if the prism is slanted or the cross sections change shape).

A commonly used SI unit for volume is the liter, and one thousand liters is the volume of a cubic meter, which was formerly termed a stere. A cubic centimeter is the same volume as a milliliter.

Volume measures: USA

Traditional US measures of volume:

• US fluid ounce, about 29.6 ml (this volume of water weighs one ounce)
• US pint = 16 ounces, or about 473 ml (this volume of water weighs one pound)
• US quart = 32 ounces or two pints, or about 946 ml
• US gallon = 128 ounces or four quarts, about 3.785 l

• UK fluid ounce, about 28.4 ml (weight of this volume of water is 28.3 g, or nearly one ounce, 28.4 g)
• UK pint = 20 fluid ounces, or about 568 ml
• UK quart = 40 ounces or two pints, or about 1.136 l
• UK gallon = 160 ounces or four quarts, or about 4.546 l

• teaspoon = 1/6 ounce
• tablespoon = 1/2 ounce or 3 teaspoons
• cup = 8 ounces or 1/2 pint

The volume, of an object, is equal to its mass divided by its average density. This is a rearrangement of the calculation of density as mass per unit volume.

Volume comparisons

To help compare different volumes, see these pages:

• 10 cm³ -
• 100 cm³ -
• 1.0 dm³ -
• 10 dm³ -
• 0.1 m³ -
• 1.0 m³ -
• 10 m³ -
• 100 m³ -
• 1,000 m³ -
• 10,000 m³ -
• 100,000 m³ -