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PERIMETER OF AN ELLIPSE

 a 3 : b 2 : 1 2 mat x 2 ^ a 2 ^ / y 2 ^ b 2 ^ / + 1 - 2 1 line 0 0 a 0 2 2 mat 0 0 0 b 2 2 mat a 2 / 0.5 a 10 red 0.2 b 2 / b 10 green 2 5 mat 4 1 sys

Major Radius

a 3 :

Minor Radius

b 2 :

Unnamed Quantity

h a b - ( 2 ^ a b + ( 2 ^ / :

Ellipse Eccentricity

e.c a 2 ^ b 2 ^ - sqrt a / :

Perfect Formula for Perimeter of an Ellipse

P.e 4 a * 1 e.c 2 ^ θ sin ( 2 ^ * - sqrt θ 0 π 2 / int * : 15.8654395892906

Method 1: Kepler Approximation

P 2 π * a b * sqrt * : 15.3905979619424

Method 2: Naive Approximation

P π a b + ( * : 15.707963267949

Method 3: Euler Approximation

P π 2 a 2 ^ b 2 ^ + ( * sqrt * : 16.0190422444141

Method 4: Peano Approximation

P π a b + ( * 3 1 h - sqrt - 2 / ( * : 15.8666459209523

Method 5: Lazy Approximation

P π 6 a * 5 / 3 b * 4 / + ( * : 16.0221225333079

Method 6: Lindner Approximation

P π a b + ( * 1 h 8 / + ( 2 ^ * : 15.8654355997101

Method 7: Hudson Approximation

P π a b + ( 4 / * 3 1 h 4 / + ( * 1 1 h 4 / - / + ( * : 15.8654395663676

Method 8: Infinite Series

P 2 a * π * 1 2 j * ( ! ( 2 j ^ j ! ( * ( 2 ^ / ( 2 ^ e.c 2 j * ^ 2 j * 1 - / * j 1 50 sum - ( * : 15.8654395892905

Method 9: Zafary Approximation

P π a b + ( * 4 π / ( h ^ * : 15.8604783620635

Method 10: Ramanujan I Approximation

P π 3 a b + ( * 3 a * b + ( a 3 b * + ( * sqrt - ( * : 15.865437575564

Method 11: Ramanujan II Approximation

P π a b + ( * 1 3 h * 10 4 3 h * - sqrt + / + ( * : 15.8654395892512 <content> <p style="font-size: 8px; font-family: Courier New;"></p> </content>

Method 1: Kepler

P 1 el 2 π * a b * sqrt * vectorize :

Method 2: Naive

P 2 el π a b + ( * vectorize :

Method 3: Euler

P 3 el π 2 a 2 ^ b 2 ^ + ( * sqrt * vectorize :

Method 4: Peano

P 4 el π a b + ( * 3 1 h - sqrt - 2 / ( * vectorize :

Method 5: Lazy

P 5 el π 6 a * 5 / 3 b * 4 / + ( * vectorize :

Method 6: Lindner

P 6 el π a b + ( * 1 h 8 / + ( 2 ^ * vectorize :

Method 7: Hudson

P 7 el π a b + ( 4 / * 3 1 h 4 / + ( * 1 1 h 4 / - / + ( * vectorize :

Method 8: Infinite Series

i 1 a length range K i el 2 a i el * π * 1 n n j 1 + 2 j * product 4 j ^ j ! ( * / ( 2 ^ 1 b 2 ^ a i el ( 2 ^ / - sqrt ( 2 j * ^ 2 j * 1 - / * j 1 100 sum - ( * : for P 8 el K : 2 1 line

Method 9: Zafary

P 9 el π a b + ( * 4 π / ( h ^ * vectorize :

Method 10: Ramanujan I

P 10 el π 3 a b + ( * 3 a * b + ( a 3 b * + ( * sqrt - ( * vectorize :

Method 11: Ramanujan II

P 11 el π a b + ( * 1 3 h * 10 4 3 h * - sqrt + / + ( * vectorize :

Perfect Method

i 1 a length range K i el 4 a i el * 1 e.c i el ( 2 ^ θ sin ( 2 ^ * - sqrt θ 0 π 2 / int * : P 12 el K : 2 1 line for 6.2832 7.6953 8.8858 9.9346 10.8828 11.7548 12.5664 13.3286 14.0496 14.7354 15.3906 16.019 16.6237 17.2072 17.7715 18.3185 18.8496 19.3661 19.8692 20.3598 20.839 21.3073 21.7656 22.2144 22.6543 23.0859 23.5095 23.9257 24.3347 24.7369 25.1327 25.5224 25.9062 26.2844 26.6573 27.025 27.3878 27.7458 28.0993 39 1 mat 6.2832 7.854 9.4248 10.9956 12.5664 14.1372 15.708 17.2788 18.8496 20.4204 21.9911 23.5619 25.1327 26.7035 28.2743 29.8451 31.4159 32.9867 34.5575 36.1283 37.6991 39.2699 40.8407 42.4115 43.9823 45.5531 47.1239 48.6947 50.2655 51.8363 53.4071 54.9779 56.5487 58.1195 59.6903 61.2611 62.8319 64.4026 65.9734 39 1 mat 6.2832 8.0095 9.9346 11.9628 14.0496 16.1723 18.3185 20.4807 22.6543 24.8365 27.025 29.2185 31.4159 33.6165 35.8197 38.025 40.232 42.4406 44.6504 46.8614 49.0732 51.286 53.4994 55.7135 57.9281 60.1432 62.3588 64.5748 66.7912 69.0079 71.2248 73.4421 75.6596 77.8773 80.0952 82.3133 84.5316 86.7501 88.9687 39 1 mat 6.2832 7.9333 9.6943 11.5261 13.4082 15.3284 17.2788 19.2538 21.2495 23.2628 25.2914 27.3334 29.3872 31.4517 33.5257 35.6085 37.6991 39.797 41.9017 44.0125 46.1292 48.2512 50.3783 52.51 54.6463 56.7867 58.9311 61.0792 63.2309 65.386 67.5442 69.7056 71.8699 74.037 76.2067 78.3791 80.5539 82.7311 84.9105 39 1 mat 6.1261 8.0111 9.896 11.781 13.6659 15.5509 17.4358 19.3208 21.2058 23.0907 24.9757 26.8606 28.7456 30.6305 32.5155 34.4004 36.2854 38.1704 40.0553 41.9403 43.8252 45.7102 47.5951 49.4801 51.365 53.25 55.135 57.0199 58.9049 60.7898 62.6748 64.5597 66.4447 68.3296 70.2146 72.0996 73.9845 75.8695 77.7544 39 1 mat 6.2832 7.9327 9.6884 11.5063 13.364 15.249 17.1535 19.0723 21.0021 22.9405 24.8856 26.8362 28.7913 30.7501 32.7121 34.6767 36.6435 38.6124 40.5829 42.5549 44.5282 46.5026 48.4781 50.4544 52.4316 54.4095 56.3881 58.3673 60.347 62.3272 64.3079 66.2889 68.2704 70.2522 72.2343 74.2167 76.1994 78.1824 80.1656 39 1 mat 6.2832 7.9327 9.6884 11.5065 13.3649 15.2508 17.1566 19.0773 21.0094 22.9504 24.8987 26.8527 28.8116 30.7745 32.7409 34.7101 36.6818 38.6557 40.6315 42.6089 44.5878 46.568 48.5494 50.5318 52.5151 54.4992 56.4841 58.4697 60.456 62.4428 64.4301 66.4179 68.4062 70.3948 72.3839 74.3733 76.363 78.353 80.3433 39 1 mat 0 6.2832 7.9302 9.6812 11.4944 13.3487 15.2315 17.1351 19.0545 20.9859 22.9269 24.8756 26.8305 28.7907 30.7551 32.7233 34.6945 36.6684 38.6447 40.623 42.603 44.5847 46.5677 48.5519 50.5373 52.5236 54.5109 56.4989 58.4877 60.4772 62.4673 64.4579 66.4491 68.4407 70.4327 72.4252 74.418 76.4112 78.4047 80.3985 39 1 mat 6.2832 7.9327 9.6884 11.5064 13.3644 15.2499 17.155 19.0747 21.0056 22.9452 24.8918 26.844 28.8009 30.7616 32.7256 34.6923 36.6614 38.6326 40.6055 42.58 44.5559 46.5329 48.5111 50.4902 52.4702 54.4509 56.4324 58.4145 60.3972 62.3804 64.364 66.3482 68.3327 70.3176 72.3028 74.2884 76.2743 78.2604 80.2468 39 1 mat 6.2832 7.9327 9.6884 11.5066 13.3649 15.2509 17.1568 19.0777 21.01 22.9514 24.9 26.8546 28.8141 30.7777 32.7448 34.7148 36.6875 38.6624 40.6393 42.6179 44.598 46.5795 48.5622 50.546 52.5308 54.5164 56.5029 58.4902 60.4781 62.4666 64.4557 66.4453 68.4354 70.4259 72.4169 74.4082 76.3999 78.392 80.3843 39 1 mat 6.2832 7.9327 9.6884 11.5066 13.3649 15.2509 17.1568 19.0777 21.01 22.9514 24.9001 26.8547 28.8142 30.7778 32.745 34.7151 36.6878 38.6628 40.6397 42.6184 44.5987 46.5803 48.5631 50.547 52.5319 54.5178 56.5044 58.4918 60.4799 62.4686 64.4579 66.4478 68.4381 70.4288 72.42 74.4116 76.4036 78.3959 80.3885 39 1 mat 1 12 mat

Ellipse Perimeter Calculation Methods Error Percentage Analysis According to Ellipse Ratio

 plot 1 el plot 2 el plot 3 el plot 4 el plot 5 el plot 6 el 6 1 sys plot 6 el plot 7 el 0 plot 9 el plot 10 el plot 11 el 6 1 sys 1.454 s *