Surface area of a right cone
The number of square units that will exactly cover the surface of a cone.
Drag the orange dots to adjust the radius and height of the cone and note how the area changes.
Recall that a cone can be broken down into two parts - the top part with slanted sides, and the circular disc making the base.
In the figure above click on "explode" to see them separately. We can find the total surface area by adding these together.
The base is a circle of radius r. The area of as circle is given by
For more, see Area of a circle.
The top section has an area given by
where r is the radius at the base, and s
is the slant height.
See also Derivation of cone area.
The slant height is the distance along the cone surface from the top to the bottom rim.
If you are given the perpendicular height, you can find the slant height using the Pythagorean Theorem. For more see
Slant height of a cone.
By adding these together we get the final formula:
This can be simplified by combining some terms, but we usually keep it this way because sometimes we
want the area of each piece separately. (See the example below).
Find the area of roof material needed to cover the conical roof shown below.
Because we are not going to cover the circular base,
we only need the area of the top, sloping part of the cone.
From the above we see that the area of of the sloping top is given by
The radius r of the cone at its base is 3ft (half the diameter),
and the slant height s is 12ft.
Substituting these into the formula we get
Things to try
- In the top figure, click 'reset' and 'hide details'
- Adjust the height and radius by dragging the two orange dots
- Calculate the surface area
- Click "Show details" to check your answer
(C) 2009 Copyright Math Open Reference. All rights reserved