Measuring the area of a circle?

[Alex Franke]

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.
[snip]
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...

I figured this had to be in some cool animation somewhere, so I went-a-googlin' and here's what I found http://demonstrations.wolfram.com/Ar...eFromSegments/

Speaking of jaws dropping to the floor, here's a hypothetical that even most adults refuse to believe: Suppose Earth is a perfect sphere, and you were able to put a rigid belt around it at the equator, nice and snug. What would happen if you loosened the belt by around, say, 6 feet? It doesn't seem like it would have that big an impact because the belt is 25,000 MILES long... but do the math with her and watch her jaw drop again!

[tseavoy]

Get a piece of string, carefully place it around the circumference of the circle, tie it, pull the loop so it is in a straight line, measure that length, divide by two, then square that quantity. That will be the area of the circle.

Tom on Marrowstone

[jkristia]

Get a piece of string, carefully place it around the circumference of the circle, tie it, pull the loop so it is in a straight line, measure that length, divide by two, then square that quantity. That will be the area of the circle.

Tom on Marrowstone

Ha - that is probably the easiest way to do it.

[Alex Franke]

Get a piece of string, carefully place it around the circumference of the circle, tie it, pull the loop so it is in a straight line, measure that length, divide by two, then square that quantity. That will be the area of the circle.

Tom on Marrowstone

I'm not sure that's correct... You mean multiply by radius?

[gsmittle]

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.

You guys just made my brain explode!

g.

[gsmittle]

[WHINE]

I have read carefully through every single response, and I swear (a lot) that NONE of this made sense…

Frustrating.

[/WHINE]

g.

[jkristia]

I'm not sure that's correct... You mean multiply by radius?

Hmmm, you are absolutely right
A = (C/(R*2))*R*R
or
A = (C/2) * R
So if you take the string, measure the length, divide it by 2 and multiply it with R. That is not obvious I think.

I keep coming back to the volume as the easiest way to 'accurately' measure the area. Make a cylinder with a Diameter of 6 (units) and a height of 1 unit. The cylinder should then contain 18.84 units ... I think .....

Anyway - this was a fun exercise

Jesper

[cgallery]

Get a piece of string, carefully place it around the circumference of the circle, tie it, pull the loop so it is in a straight line, measure that length, divide by two, then square that quantity. That will be the area of the circle.

Tom on Marrowstone

I think we have a winner.

[phi1l]

[WHINE]

I have read carefully through every single response, and I swear (a lot) that NONE of this made sense…

Frustrating.

[/WHINE]

g.

Don't worry about it, g, I'm sure some of these guys posting have has multiple semesters of calculus & differential equation & are just presenting here an abridged version of what was in their course books with out much descriptive narrative that helped them understand the concepts.

[Alex Franke]

Yeah, gsmittle -- I don't know what I'm writing about either. I just picked the 3rd word from every 14th page of the 15 math books I've owed for the past 9 years or so, and strung 'em all 2-gether. I bet I could go on forever like that.

[phi1l]

Yeah, gsmittle -- I don't know what I'm writing about either. I just picked the 3rd word from every 14th page of the 15 math books I've owed for the past 9 years or so, and strung 'em all 2-gether. I bet I could go on forever like that.

ROFLOL I sort of half suspected as much

[Uncle Cracker]

I'm inclined to just take Euclid's word for it...

[cabinetman]

Suppose Earth is a perfect sphere, and you were able to put a rigid belt around it at the equator, nice and snug. What would happen if you loosened the belt by around, say, 6 feet? It doesn't seem like it would have that big an impact because the belt is 25,000 MILES long... but do the math with her and watch her jaw drop again!

Where are you going to find a belt that long? Is there a Belts-R-Us store?
.

[tseavoy]

Get a piece of string, carefully place it around the circumference of the circle, tie it, pull the loop so it is in a straight line, measure that length, divide by two, then square that quantity. That will be the area of the circle.

Tom on Marrowstone

Of course that is wrong. The area of that square will be pi/4 that of a circle, having the same perimeter which begs the question

Mea Culpa

Tom

[radhak]

Get a piece of string, carefully place it around the circumference of the circle, tie it, pull the loop so it is in a straight line, measure that length, divide by two, then square that quantity. That will be the area of the circle.

Tom on Marrowstone

I would correct it to :
Get a piece of string, carefully place it around the circumference of the circle, tie it, pull the loop so it is in a straight line, measure that length and multiply by the radius. IOW, multiply the radius with half-the-circumference. That will be the area of the circle.

Neat way, but it pre-supposes the formulae for the circumference and the area using Pi (ie, it assumes A = Pi*R*R and C = 2*Pi*R to be true). Not easy with an argumentative 10-year-old .