
Name: frothy60
Who is asking: Student
Level: Middle
Question:
I don't understand how to find the surface area of a cone. The height is 12cm and the radius is 5cm. Thanks
Hi,
There are two possible answers:
Does the 'cone' include the slanting surface AND the base (circle)
or just the slanting surface? I will describe the slanting surface
and you can ADD the area of the circle (of radius 5cm) if you need to.

For the slanting surface: imagine you cut down the cone from the point
(apex) straight to the base and open it up onto a plane.
You will have piece of a larger circle (radius the slant height).
You will get the slantheight (the length of the cut) as the
hypotenuse of a right triangle with sides: radius (5cm) and height (12cm).
With Pythagoris' Theorem, you calculate the slant height L.

Now you have a piece of a large circle with radius L.

The next question is: what fraction of this large circle to
you have. You will then take the area of the entire large circle
and multiply by this fraction to get the area of the surface of
the cone.
How do we measure 'fractions' of a circle? One way is the angle
(in degrees) divided by the angle of the whole (360). A second
way, which will work better here, is the length of the arc of the
piece you want over the length of the whole circumference of the
circle. The whole circumference is 2 Pi L.
The length of the arc you want is the same as the length
of the circle around the base of the cone which you cut: 2 Pi (5).

You can now calculate the entire area as Pi L^{2} x ( 2 Pi (5) )/ (2 Pi L).
This will simplify, of course and when you substitute L, you will have
an answer.
Walter Whiteley

