Question

A cone has base area `363` `cm^2`. A parallel slice `5` `cm` from the vertex has area `25` `cm^2`. What is the height of the cone?

Answer 1

Height of cone `~= 19.05 cm`

Explanation:

Consider the circle at the base of the cone. We are told that this has an area of `363 cm^2.`

The area of circle is given by `pi r^2`, where r is the radius,
Hence the radius of the circle at the base `r(1)` of the cone is:

`r(1) = sqrt(363/pi) ~= 19.05/sqrt(pi)`

Now consider the circle formed by the parallel slice through the cone `5cm` from the vertex. We are told that this has an area of `25cm^2.`Hence the radius of this circle `r(2)` is:

`r(2) = sqrt(25/pi) = 5/sqrt(pi)`

Finally consider a vertical slice through the vertex of the cone perpendicular to the base. This has height (h), the height of the cone, and forms two similar triangles, one with sides h and the radius of the base `r(1)` and the other with sides `5cm` and the radius of the circle made by the parallel slice `r(2)`.

Since these triangles are similar, their corresponding sides are proportional. Hence:

`h/(r(1)) = 5/(r(2))`

`h = (5 r(1)) / (r(2))`

Substituting for `r(1) and r(2)`

`h ~= 5 * (19.05/ sqrt(pi)) / (5 / sqrt(pi))`

`h ~= (5 * 19.05 sqrt(pi)) / (5 sqrt(pi))`

`h ~= 19.05cm`