For a right angled isosceles triangle of short sides 10km the usual flat Pythagoras theorem gives a hypotenuse of 10Ö2km. Using the Pythagoras on a sphere we have the length of the hypotenuse equal to

6000cos^-1

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cos

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600

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The percentage error in this case is 0.00002%.

For the last part of the problem we have the following results.

x (km)	Curved BC	Flat BC	Actual error	% error
10	14.14213	14.14214	0.000003	0.00002%
100	141.418	141.421	0.003	0.002%
200	282.82	282.84	0.03	0.009%
300	424.2	424.3	0.1	0.02%
400	565.5	565.7	0.2	0.04%
500	706.7	707.1	0.4	0.06%
656.2515676	927	928	0.9	0.1000000%
1000	1411	1414	3.3	0.2%
5000	6610	7071	461.3	7%