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Question 179127: why does the volume of a sphere does have 4 in it's formula?
Answer by Mathtut(3670)   (Show Source):
You can put this solution on YOUR website! that is a pretty complicated question:
:
the derivation of the Volume of a sphere involves Calculus.
:
The answer is not easy to illustrate. here is a partial explanation
:
Let's start with the surface area. Take a sphere of radius R, and
imagine constructing a cylindrical box with radius R and height 2R.
You can see that the sphere will fit snugly inside this box.
Archimedes, the Greek mathematician, proved a surprising fact: the
surface area of the sphere is exactly the same as the lateral surface
area of the cylinder (that is, the surface area not including the two
circular ends). You can calculate the lateral surface area of the
cylinder and you will see that it is 4*pi*R^2. The following item in
the Dr. Math Archives describes what Archimedes did to prove this
result:
In brief, you can imagine drawing a tiny triangle on
the surface of the sphere and connecting its corners to the center of
the sphere. You have made a very narrow pyramid. The volume of a
pyramid is 1/3 times the area of the base times the height. Thus the
volume of this pyramid is 1/3 times the radius of the sphere, times
the area of that little triangle.
Now, imagine that you cover the sphere with tiny triangles, and thus
cut the sphere into millions of narrow pyramids. The total volume of
the pyramids is 1/3 times the radius of the sphere times the sum of
the areas of the tiny triangles. In other words, the volume of the
sphere is 1/3 times R times the surface area of the sphere!
V = (1/3)R * 4*pi*R^2
= (4/3)pi*R^3
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