# Calculating the annulus of a sphere with a differential change in theta

Consider the following object:

I want to calculate the area of the annulus. The annulus is within the region of $$\theta$$ and $$\theta + d\theta$$

The answer of the area of the annulus is apparently $$2\pi sin(\theta)d\theta$$ but I am unsure why this is. Any explanation would be useful

• Spherical coordinates are the way to go. Once you understand the definition you will be able to set up the integral and evaluate it quickly. – Charlie Frohman Feb 18 at 12:20

If you cut the annulus you get a rectangular ribbon, with length $$2\pi\sin\theta$$ and height $$d\theta$$. Its area is then $$2\pi\sin\theta d\theta$$.