Statistics

Problem Statement

The Greek mathematician Archimedes of Syracuse (287-212 B.C.) was most likely the first in finding a way to systematically (iteratively) approximate the value of Pi, the ratio between the perimeter and the diameter of a circle.

He inscribed and circumscribed regular polygons in/around a circle and calculated their perimeter. These are lower (inscribed polygon) and upper (circumscribed polygon) bounds for the perimeter of the circle, and therefore can be used to calculate lower and upper bounds for Pi. He then found a formula for directly calculating the perimeter of inscribed and circumscribed polygons with 2*n sides only using the values previously calculated for polygons with n sides.

You now have a slightly simpler task, considering only inscribed polygons. You are given an int numSides, the number of sides of a regular polygon that is inscribed in a circle. Return a double, the approximated value for Pi when the perimeter of that polygon is used as an approximation for the perimeter of the circle.

[Image showing a circle with an 8-sided regular polygon inscribed]

Definition

Class:

Archimedes

Method:

approximatePi

Parameters:

int

Returns:

double

Method signature:

double approximatePi(int numSides)

(be sure your method is public)

Notes

• Perimeter of a circle: 2 * PI * radius
• Perimeter of a n-sided regular polygon: n * sidelength
• Reminder how doubles are evaluated:If your result is within 1e-9 of the expected result, your solution will be evaluated as correct.If your result is between (1+1e-9) * expected and (1-1e-9) * expected, it will be evaluated as correct.

Constraints

• numSides will be between 3 and 100000, inclusive.

Examples

1. 3

   Returns: 2.598076211353316

2. 8

   Returns: 3.0614674589207183

3. 17280

   Returns: 3.1415926362832276

4. 4

   Returns: 2.82842712474619

5. 5

   Returns: 2.938926261462366

6. 6

   Returns: 2.9999999999999996

7. 7

   Returns: 3.037186173822907

8. 8

   Returns: 3.0614674589207183

9. 99999

   Returns: 3.1415926530730114

10. 100000

    Returns: 3.1415926530730216

11. 48

    Returns: 3.1393502030468667

12. 11

    Returns: 3.0990581252557265

13. 83

    Returns: 3.1408425675666343

14. 80

    Returns: 3.140785260725489

15. 68

    Returns: 3.140475187910292

16. 349

    Returns: 3.1415502262546897

17. 308

    Returns: 3.141538178911329

18. 558

    Returns: 3.141576056601735

19. 700

    Returns: 3.141582107247802

20. 129

    Returns: 3.141282121798652

21. 3523

    Returns: 3.141592237225897

22. 1162

    Returns: 3.141588826347538

23. 1540

    Returns: 3.141590474591773

24. 9361

    Returns: 3.1415925946167045

25. 7638

    Returns: 3.1415925650091037

26. 57303

    Returns: 3.1415926520160147

27. 15275

    Returns: 3.141592631441721

28. 80110

    Returns: 3.1415926527845537

29. 21281

    Returns: 3.14159264217904

30. 19060

    Returns: 3.141592639364783

31. 100000

    Returns: 3.1415926530730216

32. 100000

    Returns: 3.1415926530730216