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"An equilateral triangle and a square are inscribed in a circle, with a side of the triangle being parallel to a side of the square. The entire figure is revolved about that altitude of the triangle which is perpendicular to a side of the square. Find the ratio of the area of the sphere to the total area of the cylinder, and the ratio of the total area of the cylinder to the total area of the cone".

I know that this question has already been asked, but I do not understand the hints provided. Maybe it is because my drawing of the given situation is not correct.

Drawing of the situation

So far I managed to relate the radius of the circle with the radius of the cylinder and its height and I got the correct answer. However, the book states that both ratios are equal, but I do not understand how to relate dimensions from the cone with other dimensions. I see that [OC] = [OF] but I don't know if I can use that.

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    $\begingroup$ Your figure appears to be correct. Can you relate various dimensions of the triangle and square to the radius of the circle, and then calculate the desired surface areas? $\endgroup$ – Blue Oct 11 '19 at 18:35
  • $\begingroup$ I managed to relate the radius of the circle with the radius of the cylinder and its height and I got the correct answer. However, the book states that both ratios are equal, but I do not understand how to relate dimensions from the cone with other dimensions. I see that [OC] = [OF] but I don't know if I can use that. $\endgroup$ – Emperor_Udan Oct 11 '19 at 18:41
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    $\begingroup$ First, since comments are easily overlooked, please edit your question to include the work you have done, so that people don't waste time telling you things you already know. ... As for the triangle/cone: How's your elementary trig? $\endgroup$ – Blue Oct 11 '19 at 18:45
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    $\begingroup$ To be clear: Show your actual work (including the correct answer that you got, to save readers the trouble of having to derive it), so that we can see what you know and how you approached the problem. $\endgroup$ – Blue Oct 11 '19 at 18:50
  • $\begingroup$ You do mean area, not volume? $\endgroup$ – Connor Harris Oct 11 '19 at 18:54
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Assume a unit circle. The area of the sphere:

$$A_{sphere} = 4\pi$$

The height of the cylinder is $\sqrt 2$ and its base radius is $\frac {1}{\sqrt2}$. Then, the total area of the cylinder

$$A_{cylinder}=2\pi\left(\frac {1}{\sqrt2}\right)^2 + 2\pi\left(\frac {1}{\sqrt2}\right)\sqrt 2 = 3\pi$$

The height of the cone is $\frac 32$ and its side lengths is $\sqrt 3$. Then, the total area is

$$A_{cone}=\pi\left( \frac{\sqrt3}{2} \right)^2 + \frac 12 2\pi \left( \frac{\sqrt3}{2} \right) \sqrt3=\frac94\pi$$

Thus, both ratios are $\frac43$.

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