Question

A sphere and a cube have equal surface areas find the ratio of radius of sphere to side ofcube

Answer 1

Surface area of sphere $= 4 \pi r^{2}$

We can rewrite it to have only $r$ on the LHS, as follows:

$SA = 4 \pi r^{2}$

$r^{2} = \dfrac{SA}{4 \pi}$

$r = \sqrt{\dfrac{SA}{4 \pi}}$

Surface area of cube $= 6a^{2}$

We can rewrite it to have only $a$ on the LHS, as follows:

$SA = 6a^{2}$

$a^{2} = \dfrac{SA}{6}$

$a = \sqrt{\dfrac{SA}{6}}$

Now they asked to find the ratio of $r$ and $s$, which is:

$=\dfrac{r}{a}$

$=\dfrac{\sqrt{\dfrac{SA}{4 \pi}}}{\sqrt{\dfrac{SA}{6}}}$

We can take the Square root outside:

$=\sqrt{\dfrac{\left(\dfrac{SA}{4 \pi}\right)}{\left(\dfrac{SA}{6}\right)}}$

The $SA$ can be cancelled as follows:

$=\sqrt{\dfrac{\left(\dfrac{1}{4 \pi}\right)}{\left(\dfrac{1}{6}\right)}}$

Upon simplifying:

$=\sqrt{\dfrac{6}{4 \pi}}$

$=\sqrt{\dfrac{3}{2 \pi}}$

Therefore, the required ratio is $\sqrt{3}:\sqrt{2\pi}$