What is the Sphere?
Geometrically, a sphere is a set of all points in space that lie at the same fixed distance from a reference point. The reference point is called the center of the sphere and a fixed distance taken from the center is called the radius of the sphere. A sphere can also result if you rotate a circle about any of its diameters.
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The volume of a Sphere
If we assume the radius of a sphere to be r, then the volume of a sphere is given by the formula:
However, the volume of the sphere was derived over two thousand years ago by the Greek mathematician and philosopher- Archimedes. At the time, calculating the volume of straight 3- D figures like a cube or cuboid was fairly easy. But calculating the areas and volumes of curved figures, such as the volume of the sphere, was quite a challenge.
If we try to derive a formula for the volume of a sphere today, we will need a deeper understanding of calculus. Interestingly, calculus was derived only in the 17th century by Isaac Newton. This means that Archimedes did something similar to integral calculus, long before it was discovered, to find the volume of a sphere.
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The main challenge in finding the volume of a sphere at the time was that the surface changes direction at every point. This is how Archimedes worked with it: –
Imagine cutting a sphere in half, so you have a hemisphere. Now, imagine this hemisphere inscribed perfectly inside a cylinder.
In the first slice, the part blue part representing the sphere will be infinitesimally small. However, as we keep moving downwards, the hemisphere’s cross-section starts to grow until the green region disappears completely in the last slice.
Now keep in mind, there can be as many slices as you like, and they can be as thin as you like. Now if we multiply the areas of all these green rings with their depths, we will find the total volume of all these green rings stacked up together (excluding the blue part).
Archimedes found that this volume of the green part added up to the volume of a cone, with base radius and height being the same as that of the cylinder.
From this, he inferred that the volume of the hemisphere was equal to the volume of cylinder minus the volume of a cone having the same base radius and height.
Now, the volume of the cylinder was already known to be r2h and the formula for the volume of the cone was known to be 13r2h.
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From this, we get the volume of the hemisphere as
r2h – 13r2h = 23r2h
In this case, the radius and height are the same, and equal to r.
So, the volume of the hemisphere is 23r3.
The volume of a sphere, therefore, will be double the volume of the hemisphere, i.e.,
The formula for the volume of a sphere is V = 4/3 πr³
The cylinder volume is πR3, the cone is a third that, so the hemisphere volume is 23πR3. Thus the sphere of radius R has volume 43πR3.
The formula for the volume of a cylinder is V=Bh or V=πr2h