Volume can be explained as “how much space does this take up” or “how much water do we need to fill this thing up.” Obviously a good starting reference point for my question here is both the cube and the rectangular prism, whose volume formula is simply length*width*height. Once we get to stuff like cones, pyramids, spheres, and cylinders, formulas start throwing people in for a loop, giving you various hoops that you have to go through. Don’t even get me started on inner tubes ([https://en.wikipedia.org/wiki/Solid_torus](https://en.wikipedia.org/wiki/Solid_torus) ) and vases ([https://en.wikipedia.org/wiki/Solid_of_revolution](https://en.wikipedia.org/wiki/Solid_of_revolution) ). In any case, how did we manage to create formulas for the volumes of solids that aren’t boxes?
In: Mathematics
Integral calculus.
The basic procedure for figuring out volumes like that, is we come up with a scheme to ‘slice it up’ into a series of parallel planar slices, or concentric circular shells, or triangular wedges, or *something.* Different shapes yield more nicely to different slicings. We then take those slices and find the “limit” as we make them thinner and thinner.
Let’s make it concrete. You could approximate the volume of a hemisphere of radius 3″, for instance, by representing it as a stack of circular pancakes. You know how many pancakes it takes to stack up to 3″ high, and you know that the bottom pancake must be 6″ across, and you have a formula for figuring out the diameters of all the other pancakes. Now, approximating the volume of the hemisphere is just a matter of adding up the surface-areas of those pancakes, and multiplying by their thickness.
The approximation isn’t exact, because what you have on the plate isn’t perfectly hemispherical, it’s kind of… terraced. The pancake stack has sort of a stair-step profile to it.
But now, you could do the same thing with thin French crepes instead of thick American pancakes. The method is exactly the same, but your pancakes are thinner, so you need more of them. And your approximation will be closer, since the stair-steps are smaller.
In intergral calculus, we imagine “what would the limit of this process be, what total volume would be approached, if we could make the pancakes approach *infinitely thin*?” And that’s how we derive an exact formula.
eta: And it’s not just 3-dimensional volumes. The formula for the area of a circle, A=πr^2, can be derived with this same technique by approximating a circle as an N-gon, slicing that N-gon into triangular wedges, and then taking the limit of areas as *they* become infinitely thin. A=πr^2 is thus shown to be a consequence of the ‘area of a triangle’ formula, A=½bh, and the ‘circumference of a circle’ formula, C=2πr.
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