# 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.

### Prism

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

### Pyramid

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

### Sphere

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

### Cylinder

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.)

### Cone

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.)

### Irregular objects

The volume of an object defined by an upper bound of $f(x,y,z)$ in the Cartesian three-space can be found using a triple integral: $\int_{a_z}^{b_z}\int_{a_y}^{b_y}\int_{a_x}^{b_x}f(x,y,z)\text{ dx dy dz}$, where $(a_z,b_z)$ are the bounds of $z$ and similar bounds are defined for $x$ and $y$.

## Problems

### Intermediate

• 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)