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?

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#