Measuring the area of a circle?

[radhak]

My daughter challenged me to prove the value of Pi (you know, the constant number with the value 22/7 or approx 3.14, that figures so much whenever we talk about circles and spheres). I said we could measure the circumference and diameter of any circle and their ratio would be Pi, constantly; formula being Circumference = Pi * D. Plan to carry that out today.

But how do I do the same with the area? If I could measure the area, then the formula Area = Pi * R * R helps, but how do I actually measure the area? Grid paper comes to mind, but can't get too accurate with that...

[cgallery]

Cut it like a pizza and sum the areas of the triangles?

The more slices, the closer you'll get to the actual number.

[jkristia]

One way that comes to mind is to measure the volume of a cylinder instead, then prove the calculated volume with the actual volume by filling it with water, then transfer the water to a rectangular container and there measure L*W*H. Of cours I might have thought of the most complicated way to solve the problem - that has happened before

Jesper

[eccentrictinkerer]

In my high school math class (ca. 1962) we used a flat-bottomed pan, like a spring-form cake pan and BB's. We covered the bottom of the pan with one layer of BB's then weighed the BB's. We then did the same to a 4" x 4" box, to get the weight of a square inch.

We then compared the weights and confirmed the math. We used the same methodology to measure the area of the United States!

Our math teacher was a genius at getting us interested in math. He challenged us to measure the distance to the Sears-Roebucks tower on Lake Street from the shore of Lake Nokomis (about 1.6 miles as I recall) using only a tape measure and a protractor. Fun stuff when you're a 16 year old geek.

[LCHIEN]

One way that comes to mind is to measure the volume of a cylinder instead, then prove the calculated volume with the actual volume by filling it with water, then transfer the water to a rectangular container and there measure L*W*H. Of cours I might have thought of the most complicated way to solve the problem - that has happened before

Jesper

I like that one. Should be easy to do. An alternative would be to transfer the cylinder contents to a measuring cup, convert ounces to cubic inches then work backwards to determine Pi given the known height and diameter of the cylinder. This makes the assumption that the measuring cup is correct. whereas Jespers is more conclusive given that the ruler is used to measure all dimensions and regardless of the errors in unit markings will still be correct.

[Alex Franke]

My idea would be very similar to cgallery's, but to introduce some calculus concept too.

Since you've already measured circumference and diameter, you can show that the area is radius * half the circumference by taking some pie wedges and stacking them (alternating the rounded part to the right and left) to form a sort-of skewed rectangular shape with bumpy sides. Then imagine/visualize what the shape would look like as you take more and more ever-thinner pie slices.

The more slices you take, the less bumpy the sides become, and the less skewed the rectangle becomes. Eventually you get a rectangle, and that area is easy to calculate because the radius (making the top and bottom of the rectangle) never changed, and each side of the rectangle is 1/2 of the circumference of the circle, stretched out.

EDIT: Oh, and to get back to pi, you've already shown than pi = circumference / diameter, so that means circumference is diameter times pi. If the area is half the circumference times pi, then with substitution you can prove that area is pi * r * r.

[LCHIEN]

Here's another way that can be done entirely with a computer rather than physical measurements. Called numerical integration.

A circle can be defined as x^2 + y^2 = R^2 where R is a constant, lets use 1. Then the area is Pi x R x R = Pi.
Its all symmetrical so we can use the area of half the circle (y>=0) and double the result, or 1/4 the circle (Y>0 abd X>0) and quadruple the result for simplicity and reduction of calculations.

So the height of the half circle at any point X is Y=SQRT(R^2-X^2) or
since we are using R=1, Y=SQRT(1-x^2).

if we approximate the quarter circle as a bunch of columns equal to the height of the circle at the center point of the column then the area of the quarter circle equals the sum of the area of the columns.

as you decrease the width and use more columns the error in area (the small triangular pieces in then upper corners) becomes smaller and smaller and the approximation becomes very close.

So sweep the value of X from 0 to R, in N steps. Assuming N is 5, Then the centers of the columns (for R=1) will be at .1, .3, .5, .7, and .9.
For N steps the first column center will be at R/(2*N) and stepped R/N to the next column center.

if you use N=10 you'll probably get an answer for Pi between 3.14 and 3.3 but as you increase N you can see the sum converging upon 3.14/4. With a N=1 million, your answer will probably be good to 4 or 5 digits. (your answer is the sum of the column areas which is equal to Pi x R^2 /4 which will be Pi/4 since we picked R=1.). As you increase N you can literally see the answer converge upon Pi/4 as the little error pieces get smaller and less significant. Remember each column area will be

R/N x Y
or
R/N x SQRT(1-X^2)
where Y = SQRT(1-X^2) which is the height of the center of the column.

There are refinements, such as calculating the area of the trapezoid made by the two sides of the column rather than the center height of a simple column. What I have described is Simpson's Rule midpoint and trapezoidal methods. (Man, had to look that up, haven't used it since 1st year college). You can read more about it in Wikipedia.

You can program with with a simple BASIC program or do it on a Excel Spreadsheet. This is a great computer excercise and involves no trigonometry functions and will give an exceedingly good value for Pi as you extend N, much better than you can do with a cup of water.
Good luck.

[tung tied]

I like the idea of measuring weight to derive the area. However, you may find it difficult to find a scale to measure the weight accurately. Maybe you could load sand into an eight or nine inch cake pan. You could possibly measure the weight of sand leveled to fill the pan. You could also mark an edge of the pan and roll it on its side from mark to mark to get the circumference.

[phi1l]

I like the idea of measuring weight to derive the area. However, you may find it difficult to find a scale to measure the weight accurately. Maybe you could load sand into an eight or nine inch cake pan. You could possibly measure the weight of sand leveled to fill the pan. You could also mark an edge of the pan and roll it on its side from mark to mark to get the circumference.

You could use decent sized pieces of MDF one circular & one rectangular to calebrate the thickness and density.

But I personally prefer the circular integration method described earlier

[KenBurris]

My head hurts !

[LCHIEN]

I did a quickie BASIC program in full double precision 64-bit floating point math for simpson's rule and I got these results (always on the high side because of the geometry):
N = 1, 10% error
N = 10 .3% error
N=100 109 ppm (parts per million e.g. .000109)
N=1000 3 ppm
N=10000 .1 PPM
N=100000 .003 ppm
N=1000000 .0001 ppm, e.g. computed pi = 3.141592653934358 good to about 11 decimal places
a million steps only takes about 44 seconds on a 7-year old laptop computer with QuickBasic

It actually convereges a bit better than I predicted off the top of my head in my previous post.
Here's the program in (gasp) Microsoft QuickBasic:
' Simple Simpson's rule calculation to find Pi
' Loring Chien 1/10/2010
DEFDBL A-Z
CONST Pi = 3.141592653589793# 'used only to check the result!
R = 1
QFactor = 4 'because i'm only using one quadrant we multiply the result area by 4
INPUT "Number of steps"; N ' can be very large since its dbl precision
TotArea = 0
FirstCenter = R / (N * 2)
STEPSIZE = R / N
FOR I = 1 TO N
X = STEPSIZE * (I - 1) + FirstCenter
Y = SQR(R ^ 2 - X ^ 2)
TotArea = TotArea + STEPSIZE * Y
NEXT I
PRINT "1st Quadrant Total Area ="; TotArea
XPi = QFactor * TotArea / (R ^ 2) 'correct for radius and quadrants
PRINT "Calculated Pi ="; XPi
ErrorPi = XPi - Pi
PRINT "Approximate Error "; ErrorPi

[jkristia]

I think the question was how to measure it, not how to calculate it.
If you want to calculate it you can say (mentioned in anothe post)

Area = pi * r * r
Circumference = pi * r * 2 so pi = C / (r*2)

so
A = (c / (r*2)) * r * r

Try and use r = 3
A = pi * 9 = 28.27
C = pi * 6 = 18.84

now
A = (18.84 / 6) * 9 = 28.26

So there you have calculated A from the measured C and R. But now you have to prove it , and for that I still can not think of a better way than measure the volume.

(I hope my calculations are correct !!)

Jesper

[LCHIEN]

I think the question was how to measure it, not how to calculate it.
If you want to calculate it you can say (mentioned in anothe post)

Area = pi * r * r
Circumference = pi * r * 2 so pi = C / (r*2)

so
A = (c / (r*2)) * r * r

Try and use r = 3
A = pi * 9 = 28.27
C = pi * 6 = 18.84

now
A = (18.84 / 6) * 9 = 28.26

So there you have calculated A from the measured C and R. But now you have to prove it , and for that I still can not think of a better way than measure the volume.

(I hope my calculations are correct !!)

Jesper

to get the value of A and C you used the actual value of Pi so that's not a valid method of solving by calculation.

In my proof, I could have no knowledge of Pi, just the theorem that A= Pi R^2. I then Used a computer to "measure" the area of the circle in increasngly accurate pieces and the summed area was then used with the R value to back calculate Pi. Granted its not a physical measurement but its very easily (i would hope) to visualize.

I had fun anyway revisiting 1st year college numerical analysis.

[radhak]

All great suggestions - and make for great discussion, even with a 10 year old.

The best thing about measuring area of a pizza-slice is that if the slice is thin enough, it would leave very little error, and of course we could always multiply the area so many times, so the effort is just in measuring one triangle.

I had not thought of it, but the idea for the volume of a cylinder is neat too - she understands the LxWxH, so it's an easy grasp.

Alex - it took me a moment to appreciate your approach, but it makes sense; I think she'll need an actual demonstration to get it.

And while as a math guy your formulae appealed to me, Loring, your diagram helped her more so : she agrees we could draw many of these rectangular boxes and if evenly sized and numbered, we could average the heights of the smallest and largest : so it helps in actual measurement too! (Hope to do this tomorrow).

I don't think I'll be able to use the weight method : I don't have any weight machine; maybe some other time.

Btw - we did the circumference today, and her jaw literally dropped when she saw that the ratio came very close to 3.14 for all sorts of sizes of circles : from the dinner plate to the the dining table . For all that they had read about Pi, a practical demo is always more forceful. I'm sure she has a far deeper understanding now.

[LCHIEN]

All great suggestions - and make for great discussion, even with a 10 year old.

The best thing about measuring area of a pizza-slice is that if the slice is thin enough, it would leave very little error, and of course we could always multiply the area so many times, so the effort is just in measuring one triangle.

I had not thought of it, but the idea for the volume of a cylinder is neat too - she understands the LxWxH, so it's an easy grasp.

Alex - it took me a moment to appreciate your approach, but it makes sense; I think she'll need an actual demonstration to get it.

And while as a math guy your formulae appealed to me, Loring, your diagram helped her more so : she agrees we could draw many of these rectangular boxes and if evenly sized and numbered, we could average the heights of the smallest and largest : so it helps in actual measurement too! (Hope to do this tomorrow).

I don't think I'll be able to use the weight method : I don't have any weight machine; maybe some other time.

Btw - we did the circumference today, and her jaw literally dropped when she saw that the ratio came very close to 3.14 for all sorts of sizes of circles : from the dinner plate to the the dining table . For all that they had read about Pi, a practical demo is always more forceful. I'm sure she has a far deeper understanding now.

I could agree that using the Simpson rule and actual physical columns as rectangles could work instead of computational integration...or demonstrate the closure of the computational integration as a first step.
As I discussed above, with only ten columns you should have the error down to .3% (or about 2 good decimal digits of Pi) which is actually fantastic. A 10 inch radius circle with 1" wide columns should yield fantastic results with 1/8" or so resoution on the column heights. - you could do N =1, 5 and 10 easily and show the convergence.

Best of luck to you and your daughter.