Problem number nine, five circles, each with an area of eight centimeters squared, overlap to form the figure shown. The area of each section where two circles overlap is one centimeter squared. What is the total area covered by the figure? Let's look at the leftmost and the rightmost circles. For example, let's look at the left circle. It's divided into two parts, the overlap and the rest. We know that the area of the overlap is exactly one, and since the area of the whole circle is eight, that means that the area of the remaining part is seven. We can do the exact same thing on the rightmost circle. The area of the overlap is one, and the area of the remaining part is seven. Now let's mark the rest of the overlaps because we know they already have area one. Now let's look at any of the other circles. If we look at the circle, it has area eight, and we know that we have two regions with overlaps of size one. So that means that the remaining part has to have area of size six. For example, this left circle, this part has to be six because we already know that there are two parts of area one, and the whole part is area eight. Similarly, the other two circles also have area six in the central region. Now we just have to add up all the numbers we see. There are a lot of repeats, so we can just use multiplication. There are two sevens, three sixes, and four ones. So that is 14 plus 18 plus 4, and that is equal to 36. So the answer is B, 36 centimeters squared.

overlapping circles

total area

non-overlapping area

circle overlap

geometry problem