"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.
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.
Best Answer
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 $\frac43$.