Area and volume > Volume

You already know that the volume of all solids that have the shape of a prism or a cylinder with base area $G$ and height $h$ is equal to $G\cdot h$ .

The volume of a cuboid is therefore $V\left(\text{cuboid}\right)=G\cdot h=l\cdot b\cdot h$ .

The volume of a straight cylinder is therefore $V\left(\text{cylinder}\right)=G\cdot h=\pi {r}^{2}\cdot h$ .

You already know that the volumeof all solids that have the shape of a pyramid or a cone with base area $G$ and height $h$ is equal to $\frac{1}{3}\cdot G\cdot h$ .

The volume of a straight coneis therefore $V\left(\text{cone}\right)=\frac{1}{3}\cdot G\cdot h=\frac{1}{3}\cdot \pi {r}^{2}\cdot h$ .

According to Cavalieri's principle this is also true for slanting prisms, cylinders
and cones, als long as $h$ is perpendicular to the base (and top).

The volume of a sphere is: $V\left(\text{sphere}\right)=\frac{4}{3}\pi {r}^{3}$ .