Volume

The volume of an object is a measure of the amount of space that it occupies. Note that volume only applies to three-dimensional figures.

Finding Volume

This section covers the methods to find volumes of common Euclidean objects.

The volume of a prism of height $h$ and base of area $b$ is $b\cdot h$ .

The volume of a pyramid of height $h$ and base of area $b$ is $\frac{bh}{3}$ .

The volume of a sphere of radius $r$ is $\frac 43 r^3\pi$ .

The volume of a cylinder of height $h$ and radius $r$ is $\pi r^2h$ . (Note that this is just a special case of the formula for a prism.)

The volume of a cone of height $h$ and radius $r$ is $\frac{\pi r^2h}{3}$ . (Note that this is just a special case of the formula for a pyramid.)

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron? (Source)

A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x.$ ) (Source)