# How to calculate the geometric center of the surface-area of a part of a Sphere?

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?

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.