# Dividing a hemisphere into two equal surface areas

I have a geometry/mathematics question that I am sure someone has already answered using algebra.

I need to know what angle of spherical cap I would need to use to divide the surface area of a sphere in two.

i.e. in the image below, what angle of theta would you need so that the top half of the sphere would be divided in two parts with equal surface area?

Image from Wikipedia

For those of you who are interested: I have points which fall on the surface of a hemi-sphere, there are more points at the pole than at the equator. I would like to quantify this effect, but I want to compare two halves of the hemisphere with equal surface area. My data are described in spherical coordinates, so it is easy for me to divide them based on polar angle...

Thanks for any help,

Rod.

p.s. If someone simply knows the angle required to do this I would be ecstatic, however, if there is some mathematical proof I can use to show it that would be even better.

• Cut the height in half. Feb 6, 2018 at 1:19
• This gives the same answer as Doug M, so it is also correct. It has a nice intuitive feeling to it - I wasn't sure a hemisphere with half the height would necessarily have the same surface area though, which is why I didn't come to this answer myself. Thanks! Feb 7, 2018 at 7:49

$\int_0^{2\pi}\int_0^{\phi} \sin\phi \ d\phi\ d\theta$
Spherical cap is $\frac 12$ the area of the hemisphere.
$\int_0^{2\pi}\int_0^{\phi} \sin\phi \ d\phi\ d\theta = \frac 12 \int_0^{2\pi}\int_0^{\frac {\pi}{2}} \sin\phi\ d \phi\ d\theta\\ -\cos\phi|_0^{\phi} = \frac 12\\ \phi = \frac {\pi}{3}$