# Find the surface area of the circumcyclinder

The surface area of a sphere with radius $r$ is $4 \pi r^2$. Find the surface area of its circumscribing cylinder.

I don't know to begin the problem. I would highly value your hints

• What is the circumcylinder? – Faraad Armwood Sep 26 '16 at 6:04
• So your question is to find the surface area for a cylinder of radius $r$? – Faraad Armwood Sep 26 '16 at 6:08
• Yes, what @user366082 was supposed to discover is the wonderful fact that Archimedes knew that the sphere and the cylinder have precisely the same surface area. – Ted Shifrin Sep 26 '16 at 6:25
• Well, it's not clear if the OP meant the lateral area or the total area. – kobe Sep 26 '16 at 6:29
• What do you mean, @user366082? – Ted Shifrin Sep 26 '16 at 6:33

Interpretation of "Circumscribing Cylinder" used in your question:

Smallest possible cyclinder which can contain a given sphere

Sphere radius = $r$

So cylinder radius = $r$

Cylinder height = $2$ $\times$ sphere radius = $2r$

Cylinder surface area is equal to $2$ times the surface area of an end circle, plus the curved rectangular surface.

Ends: $2 \times \pi r^2$

Rectangle: $2r \times 2\pi r$

Surface area = $2\pi r^2 + 4\pi r^2 = 6\pi r^2$