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I have a regular sphere, $V=\pi r^3\frac4 3$ and $A=4\pi r^2$. Now I want to seperate it into four slices, with equal amounts of surface area - not counting the sliced area.

Or in other words, I want to find the distance from the bottom of half a sphere, where above and below have the same surface area.

I started by making a formular that describes the circumference of a slice, depending on the distance from center: $s=\pi \sqrt{(r^2-x^2)}$. So at the bottom, the circumference is $\pi r^2$, and at the top $0$. Makes sense. Now my approach was to integrate that from 0 to v after x, and set this equal to $A/4 = \pi r^2$. So $r^2 = \int_0^v \sqrt{(r^2-x^2)} dx$. This does not however give any meaningful solution it seems. What is wrong here?

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The area of a spherical cap is $\pi rh$, where $h$ is the distance from top to bottom of the cap. Hence you must simply divide the diameter of the sphere into four equal parts.

To understand why your computation doesn't work, see here.

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