There is a funny thing about the area of a sphere. If you look at the moon, you see a certain cross-sectional area. But the actual area of the moon is exactly four times what you are looking at. It's a funny thing, that factor of four.
If you know the area of a sphere, the volume is "trivial", as the mathematicians like to say. From the formula for the volume of a cone, 1/3(base)x(height), you get the volume of the sphere. The surface of the sphere is simply the "base" of a generalized "cone". Archimedes famously determined the volume of a sphere by a very ingenious and very different argument involving a cone inscribed in a cylinder. I don't know if he knew about the ratio of 4:1, but that's another pathway to the volume formula.
Of course it works both ways: if you know the volume of a sphere, as Archimedes did, you can back it up to get the formula for area. Again, I don't know if that's what Archimedes did. It would be nice to find out.
But without solving first for volume, is there an obvious way you can derive that special 4:1 ratio? It turns out you can indeed argue it from geometry without too much difficulty. It starts off looking a lot like Archimedes' argument: first, you incribe a sphere in a cylinder. And then you take a disc-shaped slice perpendicular to the axis of the cylinder. This is just like Archimedes so far. But then, to calculate volume, Archimedes inscribes a double-cone inside the cylinder, and it gets pretty intricate. For the surface area, you don't need the cone. You just compare the areas of the cyldrical section and the spherical section, and it's easy to see they are equal. From this, you immediately get the surface of the sphere.
For a long time, I've had the idea that this ratio had some kind of cosmic connection with physics. I don't remember where I got this idea, but think it might have been a pure accident. See, there's a funny thing about the apparent size of the sun. The apparent diameter of the sun, as viewed from the earth, is pretty close to one hundredth of a radian. That means if you hold your hand out 50 centimeters in front of your eye, and stick out your pinkie finger, the sun will span about half a centimeter as measured across your fingernail. One percent.
You have to be careful not to compare apples to oranges, or in this case radii to diameters. Your arm is a radius and the sun is a diameter, so you the true ratio is of course 1:200. Since area goes as the square of the linear dimension, that means the sun occupies one part in forty thousand as compared to...the area of the sky? No, because we don't yet know the area of the dome. We're really comparing flat circles, which would be the equivalent flat area of the sky. The beauty of the 4:1 business is that we can immediately see that the area of the dome is exactly twice the area of the corresponding disk; so it follows that the ratio of the sun's area to the total area of the sky is 1:80,000 which is a pretty cool result.
Here's where it gets cosmically weird. The dome we see is exactly one half of the actual "sky", because there is just as much sky on the other half of the world. So the area of the sun to the total sky becomes 1:160,000 or exactly half of the visible ratio.
The funny thing is that number 160,000 happens to be a perfect fourth power: specifically, it is 20^4. You know it's not that unusual to hit a perfect square. Perfect cubes are not that common. Perfect fourth powers are pretty rare...obviously, there are only 19 of them smaller than 160,000. But so what?
Fourth powers are not only a bit rare among natural numbers, but even more rare in physical laws. Most laws of physics have squares in them, but there is a law of thermodyanamics that says a black body radiates heat according to the fourth power of absolute temperature. So if you make something twice as hot, it radiates not twice as much heat, not four times as much heat (square law), but sixteen times as much.
Now we're going to put it all together. The temperature of the earth is around 300 degrees Kelvin (absolute scale). The earth is therefore radiating heat into the vastness of space according to the Laws of Thermodynamics. Unless that heat is being replaced by an equivalent source, the earth must therefore be cooling down. The fact that we aren't cooling down tells us how much heat we are aborbing from outer space.
If outer space consisted of a gigantic dark sphere at a temperature of 300 degrees Kelvin, we would obviously be in thermal equilibrium. We would be radiating heat out to the giant sphere, and it would be radiating heat back to us. Both bodies would be radiating heat at the same rate. But in fact there is no giant sphere out there, just the endless vacuum. So we are getting nothing back.
Except for this tiny patch of the sky occupied by the sun. Since it is only one part in 160,000 of the total sky, and it is obviously doing the whole job that our hypothetical giant sphere was otherwise doing, it must be giving off power at a rate 160,000 times as great as the black body of 300 degrees. But in that case, we know how hot the sun is! Knowing that a black body radiates heat according to the fourth power of absolute temperature, we take the fourth root of 160,000 and find that the sun is exactly 20 times as hot as the earth, or pretty close to 6000 degrees Kelvin.
And that's damn close to the actual temperature of the sun.