How can we say that the area of any circle or circular based shape is finite?

Question

I am saying this because the area of the circle is pi * radius * radius. We know that pi is a never ending value. So, if some one says I need a circle of 3 metre square area to make a rim of the wheel. How can I make so? The value of pi is non-terminating - not a practical value. Also, does it mean that area of a square is finite but area of circle is not finite. But, if area of circle is not finite, then how can I see/visualize it.

Answer 1

Non terminating doesn't mean infinite. pi is a finite number of a value between 3 and 3 1/4. The decimal expansion is infinite but the value is finite. pi is between 3 and 4. Go one more place value and we see pi is between 3.1 and 3.2. Go another place value and we see pi is between 3.14 and 3.15, and so on forever. The precision of pi is infinite but it's value is simply a finite number like any other.

To get an area of 3 square meters you do:

$ A = \pi*r^2 = 3 $

$r^2 = \frac{3}{\pi} $

$ r = \sqrt{\frac{3}{\pi}}$

That number is a finite number. According to my calculator it is approximately 0.97720502380583984317276924567669... Like $\pi$, this number is non-terminating so I can't express it exactly. So I'll say r = 0.977 and that is within a millimeter of what we need. If I want to measure it within the diameter of a helium atom I round it off to 10 place values. So r = 0.977205024 meters minus about 1/5 of a helium atom.

Answer 2

All space (area) is infinite. Measurements express our understanding of such via finite rounding. That is to say, in measuring we express a point as finite - when in fact a point has an area made up of an infinite number of "points."

Answer 3

A shape that is completely inside of another shape has a smaller area. A circle of radius 1 centered at the origin may be placed inside of a square with corners at (-1,-1), (1,-1), (-1,1), (1,1). The area of such a square is 4, and so the area of the circle must be less than 4.