Where did the formula for a sphere's volume come from?
[Sec 2 & onwards Mathematics - Volume of Spheres]
You might come across some strange formulae in Maths which your teacher does not explain in detail. For example: The volume of a pyramid, (⅓ x base area x height) is easily visualised by taking a simple cube and letting all 8 corners join at the centre to make 6 pyramids, however it's not always obvious at first glance.
What I'd like to cover today, is a particular formula that has captured the fascination and confusion of many students across the world - The volume of a sphere.
As you might see from the video, the annulus that forms when the cylinder subtracts the cone's area happens to have the same area as the cross-section of the hemisphere at that height. It's this critical observation that allows us to understand that since a hemisphere's area can be calculated using ⅔ of the cylinder, simply multiplying it by 2 gives you the entire sphere. That's why the sphere formula uses 4/3 x pi x r^3.
It's nice to know that having a weird cutout of the remaining cylinder (after shaving off the cone) gives you this weird groove that has the same area as the hemisphere. The connection is not obvious at all, yet the mathematics stands: Both can be defined as pi x (r^2 - h^2).
The next time you see a sphere and are forced to use the formula, think back on Cavalieri's Principle. The premise behind it is knowing a cylinder's volume, as well as a cone's volume.
Now, here's a question to keep you thinking a little further; We know, that a cylinder's volume is simply the circle base's area x height. But how do we prove that a cone's volume is ⅓ of this value? Is it really safe to believe what the textbook says, without ever trying to figure out the actual reason why?
Knowing the distinctions, the origins and the proofs behind the formulae is the difference between a simple "O-Level Maths: A2" standard to an exam setter's standard. Rather than pre-emptively predict what the exam will have, become an exam setter yourself. If you understand what the exam setter understands, you'll be able to notice (and better appreciate) what they intend for you to demonstrate with your workings.