Wednesday, May 25, 2011

Approximating π

The ancient fraction for estimating π, 22/7, is only accurate to two decimal places! Terrible, right? Well, that's about 0.04% too big. The common estimate of π to 5 digits (3.14159) is about 99.9999% of the true value. The fraction 355/113 is 0.0000085% too big. How terrible is that?

If you're trying to estimate the number of square feet in a circle with a radius of 100 feet, the fraction 22/7 adds 12.6 square feet to the true total of about 31,416 square feet. Using π to five digits to approximate the area is less than four square inches short of the true total. Using the ratio 355/113 instead of π to approximate the area overestimates the area by little more than a third of a square inch.

Say you're trying to cut a length of rope to go around a sphere the size of the Earth, and you know that the mean radius of the Earth is 6,371 km. If you use the ratio 355/113 to estimate π, then the 20,015-kilometer rope that you cut will be about a meter and a half too long. Measuring a length of rope that long is probably a bigger problem than the error in 355/113 as an approximation of π.

Wikipedia says:
...the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error of less than one millimetre, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.

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