It is very simple to provide a concrete example of how to find the circumference of a circle using its diameter. However, when I try to think of a similar exercise for area of a circle, I really can’t think of one. Is there a way to prove the formula for for area of circles with a concrete, or hands-on, example?
Yes, use a double integral in radial coordinates, it should look something like:
S [0 to 2*pi] S [0 to R] r d® d(theta)
to get pi* R^2, where S is the integral sign.
What I really am looking for is a physical way of showing this. For Circumference, I can roll a cylinder and measure how far it goes, then divide this by Pi, and it always equals the diameter.
emacknight, your explanation is a little above what I can understand. Anyway to bring it down to my level?
You can draw a circle on graph paper, then count the number of squares that you circumscribe, divide by pi and take the square root, and compare what you get with the radius.
Take a cylinder of the size of circle you want to measure. Wrap a string around it and tie the string.
Slip the string off the end of the circle and lay it down on graph paper in the shape of a square, lining the string up on the graph paper.
Then count the squares.
This is incorrecect. A square with a circumference of 4 has 1 square unit of area. A circle with a circumference of 4 has an area of 5.092.
you could fill a cylinder of known radius and height with a liquid of known density, then weigh the liquid.
Augh, what a dumbass. I used the diameter, not the radius. A circle with a circumference of 4 has an area of 1.27… square units.
Fill a cylinder of radius r to depth r with water.
The volume of the water will be Pi*r^3, the area of the circle multiplied by r.
Easy. Cut the circle in question out of plywood. Cut one square centimeter out of the same plywood. Measure the circumference in centimeters. Now weigh the circle and the square centimeter, and divide the weight of the circle by the weight of the square centimeter – that’s the area of the circle in square centimeters.
I just had the thought that you could use BBs or something similar to measure areas. You’d need a bunch of BBs, peas, or something else more or less identically sized, and a circle and square with rims to contain the BBs. Just fit as many BB’s as possible into the circle without stacking them on top of each other, and count them. Do likewise for the square (ideally it would have sides a convenient mulitiple of the circle’s radius), and you can compare areas. You’ll abviously have some error in the area measurement, due to all the gaps around the edges, but shouldn’t need a very big circle to fit a hunded BB’s into, which I think would be enough to show the general idea, and probably as many as you’d want to count.
Let us know how it goes!
I doubt it. Everything will come down to an approximation because it is a calculus problem–you’re dealing with changes that you won’t be able to get using any physical medium.
For another form of approximation, you could do what the Japanese did: Take a circle and cut into pie shapes. Rearrange the shapes to create a crude rectangle. Get the area of the rectangle.
Now, repeat this with thinner slices of pie and you will get an even closer approximation. When you can cut an infinitely thin slice of pie, you will get the exact value of pi. See why it can’t be done physically? You can only approximate.
Another method has to do with approximating it probabilistically by dropping a needle on a grid. IIRC, it is called Buffon’s Needle Problem.
You can read about these in Becker’s A History of Pi. Or maybe the author is Beckman? Yeah, I think it’s Beckman. Anyway, it is a really good book. Check it out!
Another way to approximate a circle is as an N-sided polygon. You know how to calculate the area of a triangle, you can show that the twelve triangles forming a dodecagon (for example) get close to the area of the circle. You could even graph how the area of an N-sided polygon asymptotically approaches the area of the containing circle.
Are you trying to prove this to yourself or is there a target audience? Children might be more likely to understand a physical approach, whereas adults might prefer something a little more cerebral.
Another way to approximate a circle is as an N-sided polygon. You know how to calculate the area of a triangle, you can show that the twelve triangles forming a dodecagon (for example) get close to the area of the circle. You could even graph how the area of an N-sided polygon asymptotically approaches the area of the containing circle.
Are you trying to prove this to yourself or is there a target audience? Children might be more likely to understand a physical approach, whereas adults might prefer something a little more cerebral.
Thank you all for your insight. Stypticus, the target audience is a group of 6th grade students. I think the approach that js_africanus brought up makes the most sense to me. It also shows clearly why we can never have an exact value for Pi. Thanks to all of you for bringing this one down to my level of understanding.
moejuck
Try this method to show the kids after you’ve established C=pi r.
- Cut a circle into N (10 or 12) slices
- Arrange the slices to simulate a rectangle - kind of like this
(---------
---------)
(--------
.
.
.
where the parentheses are the curved ends of the slices.
-
The area of the circle equals the area of the pseudorectangle
-
The width of the rectangle is very close to r (because each slice goes from the center to the outside of the circle) and the length of the rectangle is close to C/2 (because the length of the thick end of each slice is nearly C/N and there are N/2 thick ends on each side of the rectangle
-
Area of the rectangle equals C/2 * r = pi r^2
Cool eh?
We do have an exact value for pi. This value can be specified to any desired degree of accuracy. If you mean we cannot use physical measurements to determine an exact value, I agree, but only because we there is always a degree of error in physical measurements. We can still approximate pi using physical measurements, and we can get pretty accurate.
I think the cut out version of demonstration is suitable for a classroom setting, if you change the material to paper, and use a good scientific scale.
Get some fairly high quality paper, with a very regular thickness, such as good printer paper, or letter stationary. Get a compass, and a very accurate balance. (A good chemistry class balance should do.)
Print a circle on sheets of paper, using a drawing program. Cut out squares tangent to the circles, using a paper cutter to get right angles. Weigh the squares, individually, and in a batch equal to the number of students involved in the demonstration. Have the students cut out the circles, with scissors, leaving the line on the circle all the way around. Collect the circles, and weight them, individually, and as a batch.
Have the students do the arithmetic for their individual pieces, and you do the arithmetic for the batch. Divide the weight of the squares by four, for the weight of a square unit. The circles should weigh pi times that number. The batch weights should be fairly accurate. If you prepare your paper in advance, you should be able to get some fairly round numbers for the actual demonstration.
Tris
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