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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?

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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*}

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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.

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  • $\begingroup$ I would represent the sphere as circles formed to make a sphere, not a cone $\endgroup$ – cvogt8 Feb 17 '16 at 13:21
  • $\begingroup$ @cvogt8 How do you get the average? $\endgroup$ – CAGT Feb 17 '16 at 13:23
  • $\begingroup$ I could use the radius multiplied by the sin of an angle from 0 to pi to get the radii of every circle and then multiply that by 2pi. (Average sin from 0 to pi) * radius = 2r\pi. So I think that gets an average circumference of 4r. $\endgroup$ – cvogt8 Feb 17 '16 at 13:40
  • $\begingroup$ @cvogt8 Look at Spherical Segments here: mathworld.wolfram.com/SphericalSegment.html $\endgroup$ – CAGT Feb 17 '16 at 18:55

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