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This is a list of volume formulas of basic shapes: [4]: 405–406 ... List of surface-area-to-volume ratios – Surface area per unit volume;
where S n − 1 (r) is an (n − 1)-sphere of radius r (being the surface of an n-ball of radius r) and dA is the area element (equivalently, the (n − 1)-dimensional volume element). The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If A n − 1 (r) is the surface area of an (n ...
The surface-area-to-volume ratio has physical dimension inverse length (L −1) and is therefore expressed in units of inverse metre (m −1) or its prefixed unit multiples and submultiples. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm 2 and a volume of 1 cm 3. The surface to volume ratio for this cube is thus
The volume and area formulas may be derived by examining the rotation of the function = = for [,], using the formulas the surface of the rotation for the area and the solid of the revolution for the volume. The area is
The formula for the volume of the -ball can be derived from this by integration. Similarly the surface area element of the ( n − 1 ) {\displaystyle (n-1)} -sphere of radius r {\displaystyle r} , which generalizes the area element of the 2 {\displaystyle 2} -sphere, is given by
The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is: [6] A = 4πr 2 (sphere), where r is the radius of the sphere.
Its volume would be multiplied by the cube of 2 and become 8 m 3. The original cube (1 m sides) has a surface area to volume ratio of 6:1. The larger (2 m sides) cube has a surface area to volume ratio of (24/8) 3:1. As the dimensions increase, the volume will continue to grow faster than the surface area. Thus the square–cube law.
The volume and area formulas were first determined in Archimedes's On the Sphere and Cylinder by the method of exhaustion. Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.