# Ask Uncle Colin: Curved Surface Areas

Dear Uncle Colin,

I’ve been struggling with this: “If the surface area of a sphere to cylinder is in the ratio 4:3 and the sphere has a radius of 3a, calculate the radius of the cylinder if the radius if the cylinder is equal to its height.” Can you help?

Can You Look Into Necessary Details of Equation Resolving

Hi, CYLINDER, and thanks for your message!

There are some intimidating $\pi$s in this, but it’s not too bad if you hold your nose about those.

Let’s start with the sphere’s surface area: that’s $4 \pi (3a)^2$, which is $36\pi a^2$.

The surface area of the cylinder then has to be $27 \pi a^2$, and that’s made up of the two ends and the top: $27 \pi a^2 = 2\left(\pi r^2\right) + \left(2\pi r\right)h$. (Those brackets aren’t at all necessary - they’re just grouping logical things together. Look, I’ll get rid of them now: $27 \pi a^2 = 2\pi r^2 + 2\pi r h$.)

Now, we know that $r=h$, so we can simplify that: $27 \pi a^2 = 4\pi r^2$.

Divide both sides by $4\pi$ to get $\frac{27}{4} a^2 = r^2$, then square root: $r = \frac{3\sqrt{3}}{2} a$.

Job done!

Hope that helps,

- Uncle Colin