The value of pi in taxicab geometry technically does not exist as any taxicab shape would consist of right angles. A circle does not contain any right angles, therefore circles do not exist in taxicab geometry (so neither does the value pi).
However, if you were to place a square around a circle with diameter 1, where each side is a tangent to the circle and then removed the corners of the square repeatedly without ever removing any or the area of the circle (see Figure 1), the value of 'pi' in that case would be 4 (as pi = diameter/circumference (or in this case perimeter)).
Figure 1
Here the problem arises, as perimeter and circumference are not exactly the same thing. Circumference only pertains to a circle (and circles do not exist in taxicab geometry) whereas perimeter pertains to any geometric shape (whether in taxicab geometry or Euclidean geometry).
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Circumference of any circle = (pi) x (diameter) = (pi) x (2 x radius) = 14 pi = 43.982 (rounded)Remember in taxicab geometry pi=4Not in this case, my friend. Taxicab geometry is non-Euclidean. How far will a taxi drive in a city around a circle of radius 7? Picture four points in the cardinal directions each 7 blocks from downtown. How far to drive from one to all three others and back? Label them N, E, W, and S (not very original, but mnemonic). From N to E, no matter which way you go as long as you are heading south and east, is 14 blocks. Continuing around is going to be 56 blocks. â–
surface area. S = 4 * pi * R2 This refers to the fluence around the surface
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Any circle's circumference divided by its diameter has a value of pi
The value of pi is as follows: 3.14159265358979323846264338327950288419716939937510 etc.