Surface Area of a Sphere in Spherical Coordinates; Concentric Rings

[AmagicalFishy]

Hey, folks.

I'm trying to derive the surface area of a sphere using only spherical coordinates—that is, starting from spherical coordinates and ending in spherical coordinates; I don't want to convert Cartesian coordinates to spherical ones or any such thing, I want to work geometrically straight from spherical coordinates. I am trying to do this by integrating concentric rings. Here's a picture of what I'm talking about:

$$\phi - \text{ is the Azimuth (note; there is one instance at the center and one near the top for illustration)}\\ \theta - \text{ is the Zenith}\\ r - \text{ is the radius of the circle currently being integrated} \\ R - \text{ is the radius of the sphere}$$

I began simply by deriving the equation for the circumference of any circle:
$$\int_0^{2\pi}\!r\ \mathrm{d}\phi$$
(The arc-length is the circle's radius multiplied by the angle; d$\phi$ is the infinitesimal angle, so the integrand is the infinitesimal arc-length.)

In a sphere, the radius r of the integrated-circles varies according to the Zenith. The radius is:
$$r = R sin{\theta}$$
Then, the circumference of any given circle within the sphere, at a height designated by θ, is:
$$\int_0^{2\pi}R sin{\theta} \ \mathrm{d}\phi$$
That is, the radius of a circle multiplied by the infinitesimal angle, from zero to two-pi, as shown above, will give you the circumference of that circle.
Now, we want the integral to sum all of the circles' circumferences of the sphere, so θ has to move from top-to-bottom. The limits of integration on θ are therefore from zero to pi, and the whole integral is:
$$\int_0^\pi \int_0^{2\pi}\! \!R sin{\theta} \ \ \mathrm{d}\phi \ \mathrm{d}\theta$$

This evaluates to: 4$\pi$R
... which is close, but wrong. If I threw in another R, life would be good, but I can't well do that without knowing why. I've thought of why I should need another R, but I can't figure anything out.

Where am I missing an R factor, and why should I be factoring it in? Everything above makes pretty intuitive sense to me, and I can't point out where or why I would add in another R. Though it's often the case that I'm making a dumb mistake.

Thanks. :)

 

[marcusl]

To stay in spherical coordinates, you need to write the differential element of area in spherical coordinates. In the $\hat\phi$ direction, the differential arc is $rd\phi$. In the $\hat\theta$ direction, the differential arc is $r\sin\theta d\theta$, as you can convince yourself by drawing a diagram or looking in a calculus book. Thus the differential area is $r^2\sin\theta d\theta d\phi$ , and the total surface area is $$r^2\int_0^{2\pi}d\phi\int_0^\pi \sin\theta d\theta.$$

 

[AmagicalFishy]

Wait.

Oh, wait, I get it!

Haha! Excellent. At one point, I was on the verge of that—then I realized that the radius of the circle being integrated varied with the sin of the zenith, and (I don't know why) decided on concentric rings.

Thank you. :)

 

[Vargo]

Here is a cool fact that is related to your question.

Wrap a cylinder around that sphere. i.e. x^2+y^2 = R^2 with z between -R and R. Any region on the sphere has the same area as the corresponding area on the cylinder. The correspondence is via a radial projection out from the z axis. So, for example, the area between latitudes would be 2pi*R^2(cos(phi1)-cos(phi2)).

 

[olivermsun]

Here is a cool fact that is related to your question.

Wrap a cylinder around that sphere. i.e. x^2+y^2 = R^2 with z between -R and R. Any region on the sphere has the same area as the corresponding area on the cylinder.

That is pretty cool!

 

[marcusl]

That was actually proven by Archimedes and published in 225 BC. He was a smart guy...