Cones
Three cones can fit inside a cylinder of the same diameter and height.
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:
\(\text{volume of a cone} = \frac{1}{3} \pi r^2 h\)
The net of a cone is a circle and a A slice of the circle, cut off by two radii.. The sector creates the curved surface of the cone.
The curved surface area of a cone can be calculated using the formula:
\(\text{curved surface area} = \pi \times r \times l\)
\(l\) is the slanted height.
The total surface area of the cone is the area of the circular base plus the area of the curved surface:
\(\text{total surface area of a cone} = \pi r^2 + \pi r l\)
Example
Calculate the volume and total surface area of the cone (to 1 decimal place).
\(\begin{array}{rcl} \text{Volume} & = & \frac{1}{3} \pi r^2 h \\ & = & \frac{1}{3} \times \pi \times 3^2 \times 4 \\ & = & 37.7~\text{cm}^3 \end{array}\)
\(\begin{array}{rcl} \text{Total surface area} & = & \pi r^2 + \pi r l \\ & = & (\pi \times 3^2) + (\pi \times 3 \times 5) \\ & = & 75.4~\text{cm}^2 \end{array}\)