# 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. $$V_{cone}=\dfrac{1}{3}V_{cylinder}=\dfrac{1}{3} \pi\cdot r^2\cdot h$$$