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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?

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

1 Answer 1

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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

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  • $\begingroup$ i still do not follow. can you please tell me how we are able to get 8 from this? Does this mean that the body diagonal is the one that is 2 units long and not the face diagonal? $\endgroup$
    – Azz Likar
    Apr 5, 2021 at 6:52
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    $\begingroup$ @AzzLikar The diagonal of the cube is the diameter of the sphere or $2$. Pythagoras gives $s^2+2s^2=4\implies6s^2=8$ $\endgroup$
    – saulspatz
    Apr 5, 2021 at 6:55
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    $\begingroup$ @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. $\endgroup$ Apr 5, 2021 at 7:01
  • $\begingroup$ @EricTowers i understand. thank you $\endgroup$
    – Azz Likar
    Apr 5, 2021 at 7:02

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