If yes, what radius (or radii) will produce that result? If no, why not?

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The answer is no.

Assume that a circle had a rational circumference C. Since C = 2*pi*r, then r = C/(2*pi) or equally r = (C/2)/pi, i.e., r is a rational number divided by pi. If you plug that value of r into the area formula you obtain

A = pi*[(C/2)/pi]*[(C/2)/pi]

=(0.25C^2)/pi

Since 0.25C^2 is a rational number, the area is a rational number divided by pi, which is irrational. Thus the circumference and area of a circle cannot both be rational numbers.

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Thank you for the nice explanation.

The short answer is : No

The longer answer:

Circumference of Circle = PI x diameter

where PI = 3.141592 ... (an irrational number)

If the diameter is a rational number then the circumference has to be irrational too because of this product.

In case the circumference is a rational number, dividing it by PI will result in an irrational number.

So they can never be both rational numbers.

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That succinctly proves that the circumference and diameter can never both be rational, but what about circumference and area?

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