# Area of curved space

1. May 4, 2008

### Niles

[SOLVED] Area of curved space

1. The problem statement, all variables and given/known data
I have the following metric, which describes a 2D, positive curved space with flat geometry (ie. a sphere):

$$ds^2\,=\,dr^2\,+\,R^2 \sin ^2 (r/R)d\theta ^2$$

Here ds is the distance between two points (r, theta) and (r + dr, theta + dtheta), R is the radius of the sphere.

I want to find the area of the sphere using this metric.

3. The attempt at a solution
Using the metric, I have found the circumference of the sphere to be 2*Pi*R (big surprise). Now I want sum up "all the circumferences" on the sphere. Is that possible?

2. May 4, 2008

### Niles

I solved it!

When finding the circumference, I must find all the other circumferences, and r is in the interval [0; Pi*R/2]. Just multiply with 2 in the end (since we only found the first half), and you're set!