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
Here’s a simple example
Take a circle, radius r, lying in the x/y plane and centred on the origin. It’s area is πr^2.
Now integrate the area as you slide it up the z axis from zero to r where the radius of the circle in the x/y plane decreases from r to zero satisfying the equation for the coordinates of points lying on a sphere, x^2 + y^2 + z^2 = r^2. you have the formula for the volume of a hemisphere.
A similar process with a linear reduction in radius from z=0 to z=h gives the volume of a cone.
And so on.
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.
With the use of calculus, by taking the integral of the shape but where numbers like height and width have been substituted with variables.
For example, if you take a right triangle where one side is on the x-axis and rotate it around an axis of the graph, you get a cone. You can use integration to find the volume of this shape. If you put variables in where the height and width of the triangle should go and solve this integral then your answer to that integral will be a general purpose solution for the volume of a cone.
https://brilliant.org/wiki/volume-cone/ here is this concept visualized.
The first we did was to just measure the volume using one of the methods you said and then taking measurement from various different sizes of the shape we can make a formula which match the measurements. However when calculus were invented it came with a tool we can use to come up with the volume of any mathematical shape. Integration is to find a formula for the area under a graph. When you integrate twice for a three dimentional object you get the volume.
…cube and the rectangular prism …
I think you mean *regular right rectangular parallelepipeds*.
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