Curvature of a Sphere and Finding the Area of a Geodesic Circle

[quark.antiquark]

I am attempting to prove the following relation between the curvature K of a sphere of radius R and the area A of a geodesic circle of radius a.

K = $$lim_{a\rightarrow 0}$$ $$\frac{\pi\cdot a^{2}-A}{a^{4}}$$ $$\frac{12}{\pi}$$

I'm off by a factor of 4 (i.e. I have 3 in the numerator instead of 12) and I think it might have something to do with my calculation of the area A. I found:

A = $$\pi$$ R$$^{2}$$sin$$^{2}$$$$\frac{a}{R}$$

Where did I go wrong?

 

[lippyka]

How can I fix this?Your formula for the area of a geodesic circle is incorrect. The correct formula for the area of a geodesic circle of radius a on a sphere of radius R is A = 2πR² sin(a/R). Therefore, your limit should be K = lim_{a\rightarrow 0} \frac{\pi\cdot a^{2}-2\pi R^{2}sin\frac{a}{R}}{a^{4}} \frac{12}{\pi}.