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Four circles of unit radius are drawn with centers (1,0), (-1,0), (0,1), and (0,-1). A circle with radius 2 is drawn with the origin as its center. What is the area of all points that are contained in an odd number of these 5 circles? (Express your answer in the form "a pi + b" or "a pi - b", where a and b are integers.)

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sqdancefan

Answer:

  4π -8

Step-by-step explanation:

Consider the attached diagram. The purple areas are those contained within 1 or 3 circles. The red areas are fully equivalent to the purple areas. The area of interest is the sum of the purple and red areas, or twice the red area.

Twice the red area is the area of the circle of radius 2 less the area of a square of diagonal 4.

For a circle of radius 2, its area is ...

  A = πr² = π·2² = 4π

For a square of diagonal 4, its area is ...

  A = (1/2)d² = (1/2)4² = 8

The area of interest is the difference of these, ...

  area of interest = circle area - square area

  = 4π -8

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