The cone: Surface area and volume

The cone is the revolution volume resulting from rotating a rectangle triangle of hypotenuse $g$ (the generatrix), low leg $r$ (which is the radius) and leg $h$ (which is the height of the cone).

Also it is possible to interpret the cone as the pyramid inscribed into a prism of circular basis.

To calculate the area or volume of a cone we only need two of the following $3$ pieces of information: height, radius, generatrix, because using Pythagoras theorem we can find the third one:

$$g^2=r^2+h^2$$

The area of the side is calculated,

$$A_{lateral}=\pi \cdot r\cdot g$$

And the entire area is:

$$A_{total}=A_{lateral}+A_{basis}=\pi \cdot r(r+g)$$

Regarding the volumes, as we have already studied in the prism and the pyramid, the volume of the cone is a third of the volume of the cylinder of equal base and height.

$$V_{cone}={\displaystyle \frac{1}{3}}{V}_{cylinder}={\displaystyle \frac{1}{3}}\pi \cdot r^2\cdot h$$