Cone basics

A right circular cone has a circular base and a single point (the apex) directly above the center.
Three lengths describe it: the base radius $r$, the vertical height $h$ from the center up to the apex, and the slant height $l$ along the side.

The radius, height, and slant height form a right triangle.
The height and radius are the two legs, and the slant height is the hypotenuse, so $l^2 = r^2 + h^2$.
This lets you find any one of the three from the other two.

Two more formulas describe the size of a cone.
The base is a circle, so its area is $\pi r^2$.
The volume is one-third of a cylinder with the same base and height: $\frac{1}{3}\pi r^2 h$.

Many questions give you one measurement indirectly.
For instance, a base area of $2304\pi$ means $\pi r^2 = 2304\pi$, so $r^2 = 2304$ and $r = 48$.
From there you can use $l^2 = r^2 + h^2$ to find the height.

A cone has a radius of $3$ and a vertical height of $4$. What is its slant height?
Use $l^2 = r^2 + h^2$: $l^2 = 3^2 + 4^2 = 9 + 16 = 25$.
So $l = \sqrt{25} = 5$.

A cone has a radius of $48$ and a slant height of $52$. What is its vertical height?
Rearrange $l^2 = r^2 + h^2$ to $h^2 = l^2 - r^2 = 52^2 - 48^2 = 2704 - 2304 = 400$.
So $h = \sqrt{400} = 20$.

A cone has a radius of $6$ and a height of $10$. What is its volume?
Use $V = \frac{1}{3}\pi r^2 h$ with $r = 6$ and $h = 10$.
So $V = \frac{1}{3}\pi (36)(10) = 120\pi$.