Cones

Three cones have the same volume as one cylinder of the same diameter and height.

Cone in a cylinder

Remember the volume of a cylinder is \pi r^2 h.

The volume of the cone is one third of the volume of the cylinder.

The formula for the volume of a cone is:

Cone with h, r and l labelled

\text{volume of a cone} = \frac{1}{3} \pi r^2 h

A cone is made from a circle and a sector of a circle. The sector creates the curved surface of the cone.

The curved surface area of a cone can be calculated using the formula:

Cone with h, r and l labelled

\text{curved surface area} = \pi \times r \times l

l is the slanted height.

The total surface area of the circular base and the curved surface is:

\text{total surface area of a cone} = \pi r^2 + \pi r l

Example

Calculate the volume and total surface area of the cone (to 1 decimal place).

Cone with diameter, 6cm, height, 4cm, and length, 5cm

\begin{array}{rcl} \text{Volume} & = & \frac{1}{3} \pi r^2 h \\ & = & \frac{1}{3} \times \pi \times 3^2 \times 4 \\ & = & 37.7~\text{cm}^3 \end{array}

\begin{array}{rcl} \text{Total surface area} & = & \pi r^2 + \pi r l \\ & = & (\pi \times 3^2) + (\pi \times 3 \times 5) \\ & = & 75.4~\text{cm}^2 \end{array}