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I'm studying for the GRE and came across the practice question quoted below. I'm having a hard time understanding the meaning of the words they're using. Could someone help me parse their language?

"The number of square units in the area of a circle '$X$' is equal to $16$ times the number of units in its circumference. What are the diameters of circles that could fit completely inside circle $X$?"

For reference, the answer is $64$, and the "explanation" is based on $\pi r^2 = 16(2\pi r).$


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up vote 4 down vote accepted

Let the diameter be $d$. Then the number of square units in the area of the circle is $(\pi/4)d^2$. This is $16\pi d$. That forces $d=64$.

Remark: Silly problem: it is unreasonable to have a numerical equality between area and circumference. Units don't match, the result has no geometric significance. "The number of square units in the area of" is a fancy way of saying "the area of."

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Thanks for responding. Is a "square unit" the same as "area of"? Or is it a square circumscribed in a circle? Where are you getting (π/4)d2 ? – b-b Jul 2 '12 at 3:50
The area of a circle of radius $r$ is, as you mentioned, $\pi r^2$. So the area of a circle of diameter $d$ is $\pi(d/2)^2$, We coudld have instead worked with $r$, and translated to diameter at the end. Yes, "the number of square units in the area of" is a fancy way of saying "the area of." – André Nicolas Jul 2 '12 at 3:53
@AndréNicolas: Of course, the phrasing "the number of square units in the area of" (and "the number of [linear] units") isn't just to be fancy; it's specifically to address your objection that units of length and area don't match, saying, in effect, "We know that. This isn't about units. Just make the numbers equal to each other." This is a perfectly reasonable algebraic exercise. Apples aren't oranges, either, but we can certainly compare numbers of each. – Blue Jul 2 '12 at 8:56

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