My daughter is in middle school (7th grade). Her beginning Algebra teacher has asked the class the following question:

The area and circumference of a circle has the same measurement. Find the radius. My wife and I have deducted that the answer could only be zero since the formulas for area and circumference are never the same except when r(radius) is equal to zero. Can you help? Thanks in advance. Scot George

Hi Scot,

There are two ways to answer.

  1. (what the teacher probably expected). We want 2*pi*r to equal pi*r2. Solving 2*pi*r = pi*r2 gives either r=0 or r=2.
  2. The correct answer. One cannot compare an area to a length because the units are not compatable -- it makes no sense to ask how many inches are in a square inch! (If you changed from feet to inches, for example, you would multiply the length by 12 inches per foot, and the area by 144 sq. inches per square foot.) The question would have to be carefully worded: first fix the unit of measurement; then determine when the pure number representing units in the circumference is itself equal to the pure number representing square units in the area.
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