# Volume of a Cone

This article helps to calculate the
*volume of a cone*. It consists of a calculator, formula, and a rearranged formula, which provides the radius given the volume.

## Formula

Archimedes discovered this formula over two thousand years ago, but students are still having problems with it even today.

## Calculator Solver

This online calculator will calculate the volume given the radius and height, however it will also find the radius if given volume and height, or find height if given radius and volume. This is an ideal calculator for a teacher to make up some interesting questions, as it provides all the parameters of the formula.

## Derivation Proof

The formula derivation proof using integration calculus is quite lengthy and therefore on a separate page. Please refer to the following link.

Volume of Cone Derivation Proof

## Given Volume Find Radius

If you have the volume and perpendicular height, then you simply rearrange the formula by using the algebraic transposition method to give you the radius.

## Given Angle

If you have either of the angles A or B, and the slant height l, then trigonometry usually finds the radius. Trigonometry works because of the right-angled triangle shaded in yellow.

Using trigonometry, cos A = adjacent / hypotenuse, therefore this formula finds the perpendicular height h from the angle A and slant height l.

You can also use the sine function to find the radius r, and the slant length l. If you have two of the three variables then you can find the third unknown variable very easily.

## Given Slant Height

If you have the slant height, then the slant height l is the hypotenuse of a right-angled triangle where one side is radius r and the other is perpendicular height h. We can therefore use Pythagoras’s Theorem to build the expression below.

Simply rearrange the expression above to find the radius, assuming you have the slant height and the perpendicular height. As long you have two of the three variables then you can rearrange the formula to find the unknown.

## Given Surface Area

Given the surface area, first find the radius using this formula, and then calculate the volume using the radius.