The Shaded Region


I found this problem online while looking for IB example problems for the Perimeter, Area, and Volume unit. Some students solved it, getting the answer of 6.4 cm. I reminded them to convert cm to mm, so they got 64 mm. I looked at their work and saw that they had used A = pi * r^2 rather than 1/2 * pi * r^2 for the semicircle. They did it again, this time getting the answer of 91 mm. I thought, “how can this circle be wider than the rectangle?” and checked their work one more time. It still came out to 91 mm.

54 cm^2 – 21.8 cm^2 = 32.2 cm^2

32.2 cm^2 = 1/2 * pi * r^2

64.4 cm^2 = pi * r^2

64.4/pi cm^2 = r^2

r = sqrt (64.4/pi) cm = 4.53 cm

d = 2r = 9.06 cm = 90.6 mm = 91 mm

I brought the problem to our math teacher meeting and asked other teachers to solve it too. We all kept getting 91 mm. Finally we decided that the problem must have a typo because the semicircle took up more area than the shaded region.

Are we missing something?

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