I’m not entirely sure what you mean by “use the year’s total number of days as a base of counting”, but it looks to me like you’re trying to find the day N such that 365/n≈π, in which case N would be the 116th day of the year, which would be April 26. Since 366/π=116.5015…, this would round up to 117, which gives us April 26 in a leap year, as well.
If we use Earth’s orbit and radians, 2π and 0 would occur on the same day. My preference would be to use perihelion for the zero point, which would mean Pi Day would coincide with aphelion. Perihelion was on January 4th this year, and will be on January 3 next year, while aphelion will be on July 4 this year and July 6 next year.
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