Errors from Rounding Pi

In honor of Pi Day, I did some calculations on the error introduced by rounding pi.  This has been done plenty of times already, but it was fun.

The key point is that rather crude approximations are still plenty accurate.  Just using 3 only leads to a 4.5% error, and the common 3.14 is good for 0.05%, which is a better tolerance than anything you're likely to do.

I also list the absolute errors when calculating circumferences of circles with various radii.  If you are calculating a circle with a radius of the observable universe, you'll need about 62 digits (first calculated in 1699) to get the error below 1 Planck length.  That ought to be enough for anybody.


 Relative error:

Digits	Value	Error
1	3	4.5070342%
2	3.1	1.3239353%
3	3.14	0.0506958%
4	3.142	-0.0129662%
5	3.1416	-0.0002338%
6	3.14159	0.0000845%
7	3.141593	-0.0000110%
8	3.1415927	-0.0000015%
9	3.14159265	0.0000001%
10	3.141592654	0.0000000%

Absolute error in calculating the circumference of a circle with the given radius:

Digits	Meter	Mile	Earth	Astronomical Unit	Light Year	Galaxy	Observable Universe
1	28.32 cm	0.46 km	1,804.17 km	42,363,900 km	17,908.56 AU	14,159.27 ly	3,993.87 Mpc
2	8.32 cm	133.87 m	529.97 km	12,444,300 km	5,260.62 AU	4,159.27 ly	1,173.19 Mpc
3	3.19 mm	5.13 m	20.29 km	476,500 km	201.44 AU	159.27 ly	44.92 Mpc
4	0.81 mm	1.31 m	5.19 km	121,900 km	51.52 AU	40.73 ly	11.49 Mpc
5	14.69 µm	2.36 cm	93.61 m	2,198.01 km	0.93 AU	0.73 ly	675,900 ly
6	5.31 µm	0.85 cm	33.81 m	793.94 km	50,208,700 km	16,781.23 AU	244,100 ly
7	0.69 µm	1.11 mm	4.41 m	103.64 km	6,554,400 km	2,190.69 AU	31,869.74 ly
8	92.82 nm	149.38 µm	0.59 m	13.89 km	878,100 km	293.5 AU	4,269.74 ly
9	7.18 nm	11.55 µm	4.57 cm	1.07 km	67,922.68 km	22.7 AU	330.26 ly
10	0.82 nm	1.32 µm	0.52 cm	122.73 m	7,761.55 km	2.59 AU	37.74 ly
11	20.41 pm	32.85 nm	130.05 µm	3.05 m	193.12 km	9,656,100 km	0.94 ly
12	0.41 pm	0.67 nm	2.63 µm	6.19 cm	3.91 km	195,600 km	1,202.95 AU
13	0.41 pm	0.67 nm	2.63 µm	6.19 cm	3.91 km	195,600 km	1,202.95 AU
14	13.52 fm	21.76 pm	86.16 nm	2.02 mm	127.94 m	6,396.77 km	39.34 AU
15	6.48 fm	10.42 pm	41.26 nm	0.97 mm	61.28 m	3,063.76 km	18.84 AU
16	0.48 fm	0.77 pm	3.04 nm	71.35 µm	4.51 m	225.6 km	1.39 AU
17	76,925,300 ym	123.8 fm	0.49 nm	11.51 µm	0.73 m	36.39 km	33,476,700 km
18	3,074,700 ym	4.95 fm	19.59 pm	0.46 µm	2.91 cm	1.45 km	1,338,100 km
19	925,300 ym	1.49 fm	5.9 pm	138.42 nm	0.88 cm	0.44 km	402,700 km
20	74,713.23 ym	120,239,000 ym	0.48 pm	11.18 nm	0.71 mm	35.34 m	32,514.03 km
21	5,286.77 ym	8,508,200 ym	33.68 fm	0.79 nm	50.02 µm	2.5 m	2,300.72 km
22	713.23 ym	1,147,800 ym	4.54 fm	106.7 pm	6.75 µm	33.74 cm	310.39 km
23	86.77 ym	139,600 ym	0.55 fm	12.98 pm	0.82 µm	4.1 cm	37.76 km
24	6.77 ym	10,889.69 ym	43,109,700 ym	1.01 pm	64.02 nm	3.2 mm	2.94 km
25	0.77 ym	1,233.65 ym	4,883,700 ym	114.68 fm	7.25 nm	362.6 µm	333.59 m
26	2,069,113,700 pl	53.82 ym	213,100 ym	5 fm	316.37 pm	15.82 µm	14.55 m
27	405,829,100 pl	10.56 ym	41,787.43 ym	0.98 fm	62.05 pm	3.1 µm	2.85 m
28	34,587,700 pl	0.9 ym	3,561.43 ym	83,626,100 ym	5.29 pm	264.42 nm	24.33 cm
29	2,536,500 pl	4,082,025,600 pl	261.17 ym	6,132,600 ym	0.39 pm	19.39 nm	1.78 cm
30	61,516.66 pl	99,001,200 pl	6.33 ym	148,700 ym	9.41 fm	0.47 nm	0.43 mm
31	61,516.66 pl	99,001,200 pl	6.33 ym	148,700 ym	9.41 fm	0.47 nm	0.43 mm
32	356.91 pl	574,400 pl	2,273,880,900 pl	862.94 ym	54,572,100 ym	2.73 pm	2.51 µm
33	356.91 pl	574,400 pl	2,273,880,900 pl	862.94 ym	54,572,100 ym	2.73 pm	2.51 µm
34	14.33 pl	23,062.27 pl	91,298,100 pl	34.65 ym	2,191,100 ym	109.56 fm	100.79 nm
35	1.96 pl	3,147.15 pl	12,458,800 pl	4.73 ym	299,000 ym	14.95 fm	13.75 nm
36	0.52 pl	835.87 pl	3,309,000 pl	1.26 ym	79,414.88 ym	3.97 fm	3.65 nm
37	0.02 pl	39.27 pl	155,400 pl	3,650,060,700 pl	3,730.65 ym	186,532,700 ym	171.61 pm
38	0 pl	0.56 pl	2,231.63 pl	52,401,000 pl	53.56 ym	2,677,900 ym	2.46 pm
39	0 pl	0.56 pl	2,231.63 pl	52,401,000 pl	53.56 ym	2,677,900 ym	2.46 pm
40	0 pl	0.03 pl	133.55 pl	3,136,000 pl	3.21 ym	160,300 ym	147.44 fm
41	0 pl	0.01 pl	24.13 pl	566,500 pl	0.58 ym	28,949.81 ym	26.63 fm
42	0 pl	0 pl	0.47 pl	11,118.95 pl	703,159,600 pl	568.22 ym	0.52 fm
43	0 pl	0 pl	0.31 pl	7,393.36 pl	467,553,800 pl	377.83 ym	347,603,600 ym
44	0 pl	0 pl	0 pl	11.57 pl	731,600 pl	0.59 ym	543,900 ym
45	0 pl	0 pl	0 pl	11.57 pl	731,600 pl	0.59 ym	543,900 ym
46	0 pl	0 pl	0 pl	6.94 pl	439,100 pl	21,957,068,900 pl	326,500 ym
47	0 pl	0 pl	0 pl	0.46 pl	29,143.95 pl	1,457,197,300 pl	21,667.11 ym
48	0 pl	0 pl	0 pl	0.09 pl	5,729.68 pl	286,484,000 pl	4,259.74 ym
49	0 pl	0 pl	0 pl	0 pl	123.89 pl	6,194,300 pl	92.1 ym
50	0 pl	0 pl	0 pl	0 pl	6.81 pl	340,700 pl	5.07 ym
51	0 pl	0 pl	0 pl	0 pl	4.89 pl	244,600 pl	3.64 ym
52	0 pl	0 pl	0 pl	0 pl	0.21 pl	10,479.35 pl	9,641,002,700 pl
53	0 pl	0 pl	0 pl	0 pl	0.02 pl	1,227.78 pl	1,129,559,800 pl
54	0 pl	0 pl	0 pl	0 pl	0 pl	57.07 pl	52,503,500 pl
55	0 pl	0 pl	0 pl	0 pl	0 pl	1.47 pl	1,349,300 pl
56	0 pl	0 pl	0 pl	0 pl	0 pl	1.47 pl	1,349,300 pl
57	0 pl	0 pl	0 pl	0 pl	0 pl	0.29 pl	266,300 pl
58	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl	2,983.86 pl
59	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl	2,401.42 pl
60	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl	247.31 pl
61	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl	21.96 pl
62	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl	0.41 pl
63	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl	0.12 pl
64	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl	0.02 pl
65	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl	0 pl