# Geometry/Volume

## Volume

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

Common equations for volume:
Shape Equation Variables
A cube: ${\displaystyle s^{3}=s\cdot s\cdot s}$ s = length of a side
A rectangular prism: ${\displaystyle l\cdot w\cdot h}$ l = length, w = width, h = height
A cylinder (circular prism): ${\displaystyle \pi r^{2}\cdot h}$ r = radius of circular face, h = height
Any prism that has a constant cross sectional area along the height: ${\displaystyle A\cdot h}$ A = area of the base, h = height
A sphere: ${\displaystyle {\frac {4}{3}}\pi r^{3}}$ r = radius of sphere
which is the integral of the Surface Area of a sphere
An ellipsoid: ${\displaystyle {\frac {4}{3}}\pi abc}$ a, b, c = semi-axes of ellipsoid
A pyramid: ${\displaystyle {\frac {1}{3}}Ah}$ A = area of the base, h = height of pyramid
A cone (circular-based pyramid): ${\displaystyle {\frac {1}{3}}\pi r^{2}h}$ 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

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

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.

Image:CavalierisPrinciple.jpg