Calculating the Area of a Circle on S^2 in the Spherical Metric

[halvizo1031]

I need help with this problem:

given a cirlce on S^2 of radius p in the spherical metric, show that its area is 2pi(1-cos p)

 

[tiny-tim]

Welcome to PF!

Hi halvizo1031! Welcome to PF!

(have a pi: π and a rho: ρ and try using the X² tag just above the Reply box )

given a cirlce on S^2 of radius p in the spherical metric, show that its area is 2pi(1-cos p)

Divide the circular region into ring-shaped slices of thickness ds, and integrate …

what do you get? ​

 

[halvizo1031]

I'm not sure I understand what you wrote.

 

[tiny-tim]

I'm not sure I understand what you wrote.

Divide the circle into rings …

the area of each ring is its thickness times its length (ie its perimeter) …

use the metric to find the length of the perimeter of each ring …

then add up the areas of all the rings

 

[halvizo1031]

ok I'll give that a try. thanks!