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 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 |
(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.
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.