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### Best Answer Jim Bob says

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.

Thank you for the nice explanation.

### Steven Brown says

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.

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

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