# Why is the answer for this problem 8 sq. units?

If a unit sphere $$(r = 1)$$ circumscribes a cube, what is the surface area of the said cube?

The answer for this problem is 8 sq. units. However, my solution is as follows:

Diameter of sphere = Diagonal of Cube (through observation of cross section) This means that through Pythagorean theorem: \begin{align}2s^2= (\sqrt{2})^2 = 2 \\ s^2 = 2\\ 6s^2 = 12 \end{align}

What is wrong with this solution? Am I missing something here?

• The first thing wrong with your "solution" is that you introduce the symbol $s$ without telling anyone what it stands for. Apr 5, 2021 at 6:56
• The second thing wrong is that from $2s^2=2$ you shouldn't get $s^2=2$. Apr 6, 2021 at 6:26

The body diagonal of a cube is actually $$s\sqrt{3}$$. The face diagonals are $$s\sqrt{2}$$.
There is a right angled triangle formed by a vertical side $$s$$, the bottom face diagonal $$s\sqrt{2}$$ and the body diagonal. You use pythagoras theorem on that triangle to get the above
• @AzzLikar The diagonal of the cube is the diameter of the sphere or $2$. Pythagoras gives $s^2+2s^2=4\implies6s^2=8$ Apr 5, 2021 at 6:55
• @AzzLikar : Of course it is the body diagonal that is length $2$. The body diagonal is the greatest distance between any two points of the cube, so the diameter of a circumscribing sphere has to be at least as long as the body diagonal. Apr 5, 2021 at 7:01