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.
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