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Headphoneus Supremus
I wanted to create an "easy" formula for approximating the circumference of ellipse at reasonable accuracy. Something people can use with pen and paper. In order to keep the numbers in the formula simple I started from the fact that 22/7 is a very good approximation of pi. If we have an ellipse with semi-major axis a and semi-minor axis b, we know the circumference C is:
C = 4a, when b = 0 (totally flat ellipse, a line segment)
C = 2𝝿a, when b = a (circle).
So, why not create a formula that gives
C = 4a, when b = 0
C ≈ 2*(22/7)*b = 44b/7, when b = a.
The formula need to move from 4a to 44b/7 as b increases from 0 to a. Linearly this happens with poor approximation so I added a higher degree term and end up with this:
This "easy" formula contains only addition, multiplication, division and raising one term to the power of 2. For example lets assume we have an ellipse with a = 13 and b = 8. We simply have
C ≈ 4*13 + 8 + 9*8*8/7*13 = 52 + 8 + 576/91 = 6036/91 ≈ 66.3
Ramanujan's formula gives 66.91180432 in comparison. The difference is about 1 %.
C = 4a, when b = 0 (totally flat ellipse, a line segment)
C = 2𝝿a, when b = a (circle).
So, why not create a formula that gives
C = 4a, when b = 0
C ≈ 2*(22/7)*b = 44b/7, when b = a.
The formula need to move from 4a to 44b/7 as b increases from 0 to a. Linearly this happens with poor approximation so I added a higher degree term and end up with this:
This "easy" formula contains only addition, multiplication, division and raising one term to the power of 2. For example lets assume we have an ellipse with a = 13 and b = 8. We simply have
C ≈ 4*13 + 8 + 9*8*8/7*13 = 52 + 8 + 576/91 = 6036/91 ≈ 66.3
Ramanujan's formula gives 66.91180432 in comparison. The difference is about 1 %.
