# Geometry/Volume

## Volume[edit | edit source]

Volume is like area expanded out into 3 dimensions. Area deals with only 2 dimensions. For volume we have to consider another dimension. Area can be thought of as how much space some drawing takes up on a flat piece of paper. Volume can be thought of as how much space an object takes up.

## Volume formulae[edit | edit source]

Common equations for volume: | ||
---|---|---|

Shape | Equation | Variables |

A cube: | s = length of a 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 spherewhich 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
. |

(The units of volume depend on the units of length - if the lengths are in meters, the volume will be in cubic **meters**, etc.)

## Pappus' Theorem[edit | edit source]

The volume of any solid whose cross sectional areas are all the same is equal to that cross sectional area times the distance the centroid(the center of gravity in a physical object) would travel through the solid.

Image:PappusCentroidTheoremExample.jpg

## Cavalieri's Principle[edit | edit source]

If two solids are contained between two parallel planes and every plane parallel to these two plane has equal cross sections through these two solids, then their volumes are equal.