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Proof for sphere
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Surface area of a sphere
To derive the formula of the surface area of a sphere, we imagine a sphere with many pyramids inside of it until the base of all the pyramids cover the entire surface area of the sphere. In the figure below, only one of such pyramid is shown
Then, do a ratio of the area of the pyramid to the volume of the pyramid
The area of the pyramid is A
The volume of the pyramid is V = (1/3) Ã— A Ã— r = (A Ã— r) / 3
So, the ratio of area to volume is A / V = A Ã· (A Ã— r) / 3 = (3 Ã— A) / (A Ã— r ) = 3 / r
Now pay careful attention to the following important stuff!
Observation # 1:
For a large number of pyramids, let say that n is such large number, the ratio of the surface area of the sphere to the volume of the sphere is the same as 3 / r
Why is that? That cannot be true! Well, here is the reason:
For n pyramids, the total area is n Ã— A
Also for n pyramids, the total volume is n Ã— V
Therefore, ratio of total area to total volume is n Ã— A / n Ã— V = A / V and we already saw before that A / V = 3 / r
Observation # 2:
Furthermore, n Ã— A_{pyramid} = A_{sphere} (The total area of the bases of all pyramids or n pyramids is approximately equal to the surface area of the sphere)
n Ã— V_{pyramid} = V_{sphere} ( The total volume of all pyramids or n pyramids is approximately equal to the volume of the sphere
Putting observation # 1 and # 2 together, we get:
Therefore, the total surface area of a sphere, call it SA is:
SA = 4 Ã— pi Ã— r^{2}
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