This brainteaser was written by Derrick Niederman. Assuming that the circumference of each circle below passes through the centers of the other two, and that the radius of each circle is 1, what is the total gray area? Resources for Teaching Math
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http://illuminations.nctm.org
Solution: 3 + 32π . At first glance, it appears that in order to find the area of the entire shape, you’ll need to find the area of the wedge‐shaped intersections in the middle. It is certainly possible to solve the problem that way, but it is easier to find a solution by drawing three lines as follows: These lines separate the total area into an equilateral triangle and three semicircles. The side lengths of the triangle are all 2, so the area of the triangle is 3 . (Divide the equilateral triangle in half along its vertical line of symmetry. The two smaller triangles can be arranged to form a rectangle with base 1 and height 3 , where 3 is found using the Pythagorean theorem or from the relationship among sides in a 30‐60‐90 triangle.) As for the semicircles, using the formula A = π r 2 , each has an area of π2 , so the total gray area equals 3 + 32π . A similar, but slightly more complicated, solution is possible by connecting the three centers of the circles to form a smaller equilateral triangle than the one shown in the figure above. (In fact, this smaller triangle is half the size of the triangle above.) Using this construction, you’ll find that the area of the “curved triangle” and football‐shaped wedges are as shown below, and then a little adding and subtracting will yield the same answer. = = π− 3
2
2π
3
−
3
2
Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics, Inc.
http://illuminations.nctm.org