# Area of square in spherical geometry

1. Dec 2, 2012

### davon806

1. The problem statement, all variables and given/known data
It is a badly drawn sphere :tongue:
By common sense,the area of the shaded region in the sphere = area of square = r^2
But can anyone show me the mathematical proof?
Moreover,does it apply to the reality?
Imagine when you bend a square sheet with length = r,does the length of curve = r after you bend it?When you bend a substance(with a small force),its molecular structure will change slightly,which means the length of side of the substance will change slightly?
I don't know whether I should post this here.If I post it at the wrong place,please move this
thread to the correct position.Thx :)

2. Relevant equations

3. The attempt at a solution

File size:
80.6 KB
Views:
159
2. Dec 2, 2012

### Dickfore

Your square is delimited by lines with constant azimuthal angle (parallels), and lines with constant polar angle (meridians) in spherical coordinates. The element of area is given by:
$$dA = R^2 \, \sin \theta \, d\theta \, d\phi$$
Therefore, you get the area by doing the multiple integral:
$$A = R^2 \, \int_{\phi_1}^{\phi_2} \int_{\theta_1}^{\theta_2} \sin \theta \, d\theta \, d\phi$$