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 which 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.)

Can you tell what the answer is? I want a clear explanation too. Thanks!!!!!!

the small circles intersect in lens-shaped areas of pi/2 - 1

each small circle has area pi

The large circle has area 4pi

The points in the lenses and the large circle outside the small circles lie in 1 or 3 circles.

4pi - 4(pi) + 4(pi/2-1) = 2pi-4

That is not right.

So, if it's not right, maybe you could let us know where we went wrong. The idea is to help here, you know.

the answer is 4pi-8 . FOR SURE!!

it is 4pi-8

correct ^^^^^

To find the area of all points contained in an odd number of the given circles, we need to consider the overlapping regions.

Let's start by examining each circle individually:

1. The first circle with a center at (1, 0) and a radius of 1 has an area of π * r^2 = π * 1^2 = π.

2. The second circle with a center at (-1, 0) and a radius of 1 also has an area of π.

3. The third circle with a center at (0, 1) and a radius of 1 has an area of π.

4. The fourth circle with a center at (0, -1) and a radius of 1 has an area of π.

Now, let's consider the overlapping regions:

5. The fifth circle with a center at the origin (0, 0) and a radius of 2 also has an area of π * r^2 = π * 2^2 = 4π.

To find the total area of the points contained in an odd number of these circles, we need to subtract the overlapping regions that are counted twice.

1. The first and second circles intersect at the origin, creating a common area that needs to be subtracted. This overlapping region has an area of π.

2. Likewise, the first and fourth circles intersect at the origin, creating another overlapping region with an area of π.

3. The second and fourth circles intersect at the origin, creating yet another overlapping region with an area of π.

4. Finally, the first, second, and fourth circles all intersect at the origin, forming one more overlapping region with an area of π.

To calculate the total area, we add the individual circle areas and subtract the overlapping regions:

Total area = (π + π + π + π + 4π) - (π + π + π + π)
= 9π - 4π
= 5π

Therefore, the area of all points contained in an odd number of these five circles is 5π, or 5π + 0.