Problem #2180

2180.

A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\frac{2}{\pi}$, as shown. What is the perimeter of this region?

[asy] unitsize(1cm); defaultpen(.8);  filldraw( circle( (0,1), 1 ), lightgray, black ); filldraw( circle( (0,-1), 1 ), lightgray, black ); filldraw( circle( (1,0), 1 ), lightgray, black ); filldraw( circle( (-1,0), 1 ), lightgray, black ); filldraw( (-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle, lightgray, black ); [/asy]

$\mathrm{(A) \ } \frac{4}{\pi}\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } \frac{8}{\pi}\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } \frac{16}{\pi}$

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Instructions for entering answers:

  • Reduce fractions to lowest terms and enter in the form 7/9.
  • Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
  • Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
  • Exponents should be entered in the form 10^10.
  • If the problem is multiple choice, enter the appropriate (capital) letter.
  • Enter points with parentheses, like so: (4,5)
  • Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

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