Area of square in spherical geometry

[davon806]

Homework Statement

Please see the attached.
It is a badly drawn sphere
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 :)

Homework Equations

The Attempt at a Solution

 

[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:
$$A = R^2 \, \int_{\phi_1}^{\phi_2} \int_{\theta_1}^{\theta_2} \sin \theta \, d\theta \, d\phi$$