Geometry/Volume

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Volume[edit]

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]

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

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(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]

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]

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