Surface Area of Sphere as Stacked Circles

Question

I know the formula is $4 \pi r^2$. I think it makes sense to say that if I represent a sphere as a stack of circles, the surface area of the sphere should be equal to the sum of the circumferences of the circles or the average circumference of the circles multiplied by the height of the sphere. Since the height of the sphere is $2r$, that means that the average circumference must be $2 \pi r$ to achieve $4 \pi r^2$. $2\pi r$ is the maximum circumference of the circles, not the average. Why is multiplying the average circumference of stacked circles by the height of the sphere to get the surface area wrong?

Answer 1

Arc length should be used instead.

Let $(x,y)=r(\cos t, \sin t)$. Then $(\dot{x},\dot{y})=r(-\sin t, \cos t)$.

\begin{align*} S &= \int_{-r}^{r} 2\pi y\sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx \\ &= 2\pi\int_{-\pi}^{\pi} y\sqrt{\dot{x}^{2}+\dot{y}^{2}} dt \\ &= 2\pi \int_{-\pi}^{\pi} r^{2} \sin t \, dt \\ &= 4\pi r^{2} \end{align*}

Answer 2

Your question: Why is multiplying the average circumference of stacked circles by the height of the sphere to get the surface area wrong?

Answer: You would get the area of a cone. There is no curvature as you go up in height, just a 45º cone.