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Make an infinite amount of lines parallel to the diameter, all the way until you get to a line tangent to the circle.

My calculations tell me this would be the sum of the infinite numbers in between the lengths 0 and 4 divided by the number of the infinite number of chords parallel to the diameter, which would be the average length of the height of the circle as a whole.

So (0 ... 4)/∞, which doesn't make sense. I can tell I need a calculus-knowledgable person to help me out here.

I tried to explain what I'm trying to get on Yahoo Answers and failed miserably.

The equation of the circle should be x^2 + y^2 = 4
So de length of a given chord parallel to the diameter is 2*sqrt(4-h^2)

Discretize the interval [-2,2] into 2N points, the average lenght of the chords should be:
1/(2N) * Sum(k=-N:N) 2*sqrt(4-(2*k/N)^2) =
1/N * Sum(k=0:N) 4*sqrt(1-(k/N)^2) =
4*Integral(0:1)sqrt(1-x^2)=
4*Pi/4 = Pi

Another question. By taking the average height of a circle and squaring it, would you get the area of the circle? If so, then the average height of a 4=d circle could be found by taking the area with pi and taking the square root. Now... assuming that the average height^2 DOES equal the area of the circle... the average height would be 3.5449077.

Before you edited your post, you said you understood qqwref. What I did was nothing other than to rephrase him in a formula form.

By definition dividing the area of an object by it's width will give you the height. That's what I did. I divided the area (pi*r^2) by the width (2pi).

Here an alternate explanation:

A circle can be described as x^2+y^2=r^2
I'm going to use r=1 to make it easier.

First, we want to find the height as a function of y. Because the height of the circle is two times the (positive) x-coordinate, we find: h=2*sqrt(1-y^2)

Next, we're going to split the circle into four vertical strips and average out the heights. The strips' x-ranges are -1 to -0.5, -0.5 to 0, 0 to 0.5 and 0.5 to 1. To get better accuracy we are going to measure the height at the middle of each strip:

Ideally we would want columns of zero with to get an exact answer. To do this, there is a trick called integration. By taking the integral from -r to r (in our case, -1 to 1) we find the average height of a radius 1 circle.

This integral comes out to pi, which means that the average height of a circle is pi*r.

By definition dividing the area of an object by it's width will give you the height. That's what I did. I divided the area (pi*r^2) by the width (2pi).

But the width was 4... unless you mean that 2pi is the average width? Sorry, I'm just trying to keep up here.
And earlier you said that the average height^2 wouldn't equal the area... right?

A circle can be described as x^2+y^2=r^2
I'm going to use r=1 to make it easier.

First, we want to find the height as a function of y. Because the height of the circle is two times the (positive) x-coordinate, we find: h=2*sqrt(1-y^2)

Next, we're going to split the circle into four vertical strips and average out the heights. The strips' x-ranges are -1 to -0.5, -0.5 to 0, 0 to 0.5 and 0.5 to 1. To get better accuracy we are going to measure the height at the middle of each strip:

Ideally we would want columns of zero with to get an exact answer. To do this, there is a trick called integration. By taking the integral from -r to r (in our case, -1 to 1) we find the average height of a radius 1 circle.

This integral comes out to pi, which means that the average height of a circle is pi*r.

Ok... I think it makes sense. That's the sort of idea I had in the first place... I'd read about the Greeks getting approximations of the area of a circle by putting many-sided polygons inside one... and they made more and more sides to it until they got a pretty accurate answer.

By definition dividing the area of an object by it's width will give you the height. That's what I did. I divided the area (pi*r^2) by the width (2pi).

But the width was 4... unless you mean that 2pi is the average width? Sorry, I'm just trying to keep up here.
And earlier you said that the average height^2 wouldn't equal the area... right

I dunno, I think it sounds reasonable that the average height*average width of any object should equal the area. Not that that makes it true, though...

I dunno, I think it sounds reasonable that the average height*average width of any object should equal the area. Not that that makes it true, though...