In this sequel to the video "Volume of a ball in n dimensions", I calculate the surface area of a sphere in R^n, using a clever trick with the Gaussian function exp(1/2 |x|^2). Along the way, we discover the coarea formula, which is the analog of polar coordinates, but in n dimensions. Finally, I show why the volume of a ball is in fact the integral of the surface area of the sphere. Enjoy!