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 wit
h 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.)
1 answer:
Check the picture 1.
The regions contained in an odd number of these 5 circles are
the pink colored region, contained only in the circle with radius 2,
and the yellow regions, each contained in the intersection of 2 small circles and the third large circle.
So what we need to determine is the "area pink + area yellow".
Consider the second picture.
Let A2 be the area of the right isosceles triangle, and A1 be half of the yellow region.
the region A2 + A1 is called a sector, and it is 1/4 of the area of a unit circle.
Thus, the overall yellow region is
The purple area is:
Area of large circle - 4*(Area of 1 small circle) + yellow region.
(because subtracting all 4 small circles means that each of the 4 separate yellow regions have been subtracted twice)
Finally, the total area is :
(units squared)
Answer:
square units
The regions contained in an odd number of these 5 circles are
the pink colored region, contained only in the circle with radius 2,
and the yellow regions, each contained in the intersection of 2 small circles and the third large circle.
So what we need to determine is the "area pink + area yellow".
Consider the second picture.
Let A2 be the area of the right isosceles triangle, and A1 be half of the yellow region.
the region A2 + A1 is called a sector, and it is 1/4 of the area of a unit circle.
Thus, the overall yellow region is
The purple area is:
Area of large circle - 4*(Area of 1 small circle) + yellow region.
(because subtracting all 4 small circles means that each of the 4 separate yellow regions have been subtracted twice)
Finally, the total area is :
Answer:
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