Suppose a sphere and a cube have equal surface areas. using r for the radius of the sphere and s for the side of a cube, enter an equation to show the relationship between r and s. if necessary, round to the nearest tenth

Accepted Solution

a cube is made of 6 squares stacked up to each other, like the one in the picture below.

now if each square has a side length of "s", then its area is s*s or s², and since we have 6 squares in the cube, its surface area is then 6s².

as far as the surface area of the sphere we already know is 4πr².

and recall that both SA are equal, thus

[tex]\bf \stackrel{\textit{SA of the square}}{6s^2}~~~~=~~~~\stackrel{\textit{SA of the sphere}}{4\pi r^2} \\\\\\ \cfrac{6s^2}{4\pi }=r^2\implies \cfrac{3s^2}{2\pi }=r^2\implies \sqrt{\cfrac{3s^2}{2\pi }}=r\\\\ -------------------------------\\\\ s^2=\cfrac{4\pi r^2}{6}\implies s^2=\cfrac{2\pi r^2}{3}\implies s=\sqrt{\cfrac{2\pi r^2}{3}}[/tex]