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In this video, the woman says that a sphere is a pretty simple object. What intrigues me is the use of a sphere for such a calculation. First of all, the sphere wouldn't be perfect as a perfect sphere doesn't exist in nature (first source of imprecision). Second, one will have to use π for calculating the volume. As π is irrational, one will have to use an approximation for it (second source imprecision). So why not using a simple cube? The shape would be closer to a cube due it's harder to make a sphere and it's volume would be calculated more precisely. Why they choose to do this calculation using a sphere?

Qmechanic
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moray95
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    Making a perfect right angle is harder than making a perfect circle. – David H Apr 14 '13 at 18:17
  • Really? I always thought that a perfect circle was the most difficult shape to make. – moray95 Apr 14 '13 at 18:28
  • Also, you're right that truly perfect spheres don't exist in nature. But, what's a greater source of error/imprecision in measurements of the figure of the Earth: using an approximation of Pi accurate only to 40 significant figures, or modeling the Earth as a simple cube? – David H Apr 14 '13 at 18:28
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    π is not really a source of imprecision. It can be calculated to any arbitrary procision. A million digits for example. – Tesseract Apr 14 '13 at 18:31
  • Related: http://physics.stackexchange.com/q/32120/2451 – Qmechanic Apr 14 '13 at 18:31
  • I'm not saying that we have to model Earth as a cube but if I was doing this calculation, I would use a cube instead of a sphere thinking that it will have better precision. Apparently not... – moray95 Apr 14 '13 at 18:31
  • Actually, circles are one of the two easiest plane figures to draw (the other being line segments). All you need is a compass (the geometry kind). See Euclid's 3rd postulate: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/post3.html – David H Apr 14 '13 at 18:40

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Because spheres are easy to make, easy to polish and they haven't got any edges, which is very much important to create an object of such a high accuracy. In fact, no one said that this is a perfect sphere. It is polished to around a few nanometers away from being perfect. And, there is no macroscopic object that is ideal to a cent percent perfection. So, the first point isn't good...

I suspect that the reason for your confusion is regarding the volume difference between both the case. The volume of a cube is very easy to determine. But, making it to such a high precision is a very big problem (as I've mentioned at first) because of its edges. As David says, right angles are very difficult.

You're right that $\pi$ can only be approximated. Still, we can use the maximum approximations as possible and we would - because, it's gonna be defined under the SI.

Waffle's Crazy Peanut
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    Its more about it being mathematically simple. If you shrunk the earth down to the size of a pool ball then it'd more of a perfect sphere than anything we've created. There's a joke in physics, but it only works for a spherical cow in a vacuum. – MaybeALlama Apr 14 '13 at 19:42