A prism is a solid with two identical parallel bases joined by flat rectangular faces — a box is a rectangular prism, a triangular tent is a triangular prism. Volume always equals base area × height. A cylinder is just a "circular prism." A cone holds exactly one-third the volume of a cylinder with the same base and height — pour a cone of water three times into a matching cylinder and it exactly fills it. The lateral surface area (the curved side only, no bases) of a cylinder unrolls into a rectangle of dimensions $2\pi r \times h$. For a cone, the curved side uses the slant height $l=\sqrt{r^2+h^2}$.
$$V_{\text{prism/cylinder}}=A_{\text{base}}\cdot h \qquad V_{\text{cone}}=\frac{1}{3}A_{\text{base}}\cdot h$$
$$LSA_{\text{cylinder}}=2\pi rh \qquad LSA_{\text{cone}}=\pi r l,\quad l=\sqrt{r^2+h^2}$$
Volume of a Cone
Find the volume of a cone with radius 3 m and height 12 m.
★★★ What Fraction of the Cylinder Does the Cone Fill?
A cone and a cylinder have the same base radius $r$ and the same height $h$. What fraction of the cylinder's volume does the cone occupy? If the cylinder holds 240 liters, how many liters does the cone hold?
