Surface Area of a Sphere

Unlike some shapes, the surface area of a sphere is not easy to calculate. All of the "easy" methods for figuring it out require the use of calculus (as in this explanation), so we'll just present the formula without proof:

The surface area of a sphere with radius $r$ is $\boxed{A=4\pi r^2}$.

Let's test your understanding! As mentioned above, it's not easy to figure out the exact factor of $4\pi$ in front, but using similarity, we can still understand the $r^2$ factor intuitively. For example, here's a qualitative question about units and scaling:

(We could also figure the previous problem out without the exact formula for surface area, just by reasoning about similar spheres.)

The moral of the story is that (surface) area is "two-dimensional", explaining the $r^{\boxed2}$ (exponent of $2$) in the surface area formula. This makes sense if you think in terms of units: possible units for radius (a "one-dimensional" length) are $m,cm,smoots,\ldots$, so some possible units for surface area (which is "two-dimensional") are $m^2,cm^2,smoots^2,\ldots$. (If you're curious about smoots, see Wikipedia.)

Now some number-crunching:

And finally:

Congratulations, you now know how to find the surface area of a sphere!