Area of circle is in square units???

Steven G

Steven G

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We all know that the area of a circle is in square units.

Since a circle is round one needs (MUST!) to explain how this can be to students. Personally, if the area is 10 sq inches, I tell students that if you cut the circle the correct way you can piece together exactly ten-square inch pieces with nothing left over. Of course I include a diagram with the explanation.

I was wondering how some of you explain this and/or think about there.
 
I can't see how to do this problem without some kind of limit, so I guess I'd use a polygon with a lot of sides and take "slices" from the center to each of the vertices. Then you would have a large number of slices, which can be piled one on top of the other to form a rough rectangle.

Or you could really weird them out and apply the Axiom of Choice and slice the circle into smaller and smaller bits until you have enough bits to make two circles... ?

-Dan
 
Don't you need infinitesimal scissors to do that?

Area is measured in units-units (like foot-pounds - obviously not an area measure). "square units" or "units squared" is just a common (and possibly confusing) way to say it when the two units are the same.
 
Great, now I'm thinking about measuring things in foot-inches. :) I guess that's kind of how it happens in lumber. A "linear foot" is a linear measure, but it implies area, based on the unnamed width of the board.
 
An odd thought: we don't have to measure area in square units; we could just as well define a unit of area called the circular inch, the area of a one-inch-diameter circle. Then we'd want to figure out what the area of a rectangle is in circular inches. In fact, it would be hard to cut a 2-inch circle into 1-inch circles, wouldn't it? Nothing would be easy to work out by counting.

I think you have to think in terms of limits, or, more elementarily, of putting a grid over the circle, counting whole squares, then subdividing the partial squares into smaller squares and counting them, and so on. A circle can't be dissected into a finite number of squares.
 
Most things that we measure the area of are not squares (not just circles). I don't see the issue.
 
No, we don't have to measure in square units. Yes, "circular units" would be a pain because circles don't "fit together", but hexagons do! We can "tessellate" an area with hexagons and use, say, "hexagonal inches" to measure it.
 
This is why non-standard analysis is heplful to intuitive explanations.
 
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