How do you find the diameter of a cone when only given the lateral area and the slant height?

Answer 1

Suppose the lateral area is A square units and the slant height is L units.

Unless you are very good at visual imagery, I suggest you try actually doing the following. You don't need to be particularly accurate but it will help you understand it better.

Cut the cone from its vertex to the base by a straight line. Open up the cone and lay it flat. This will form a sector of a circle with radius L units. The area of this sector is the lateral area of the cone. Also the arc of the sector formed the circumference of the base of the cone.

Suppose this sector subtends an angle of x degrees at its centre (what used to be the apex of the cone). That is, the sector is (x/360) of a whole circle.

The area of sector = pi*L^2*(x/360) square units = lateral area = A

Rearranging gives x = 360*A/[pi*L^2]

Then length of arc = 2*pi*L*(x/360)

Now, this length forms the circular base of the cone, so its diameter is L*(x/360).

Therefore, substituting for x gives, d = L*a/[pi*L^2} = a/pi*L