Area of square in spherical geometry

1. Dec 2, 2012

davon806

1. The problem statement, all variables and given/known data
Please see the attached.
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

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