# 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.

## Contents

## Finding Volume

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

### Prism

The volume of a prism of height and base of area is .

### Pyramid

The volume of a pyramid of height and base of area is .

### Sphere

The volume of a sphere of radius is .

### Cylinder

The volume of a cylinder of height and radius is . (Note that this is just a special case of the formula for a prism.)

### Cone

The volume of a cone of height and radius is . (Note that this is just a special case of the formula for a pyramid.)

### Parallelepiped

The volume of a parallelepiped spanned by vectors is

### Irregular objects

The volume of an object defined by an upper bound of in the Cartesian three-space can be found using a triple integral: , where are the bounds of and similar bounds are defined for and .

## Problems

### Introductory

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

### Intermediate

- A tripod has three legs each of length 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 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then can be written in the form where and are positive integers and is not divisible by the square of any prime. Find (The notation denotes the greatest integer that is less than or equal to ) (Source)