# What is the average height of a circle?

#### Nukoca

##### Member
Assume the diameter(height) of a circle is four.

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.

Help?

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#### spdqbr

##### Member
This sounds like a job for calculus! Particularly the mean vale theorem.

Really, have you studied limits at all?

#### fanwuq

##### Member
Edit:
The average height would be the area of the circle divided by the horizontal amount of space the circle takes up, right? So (2*2*pi)/4 = pi.
OK. So the average height is not at the midpoint. Guess that was a horrible assumption.

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#### qqwref

##### Member
The average height would be the area of the circle divided by the horizontal amount of space the circle takes up, right? So (2*2*pi)/4 = pi.

#### Nukoca

##### Member
Really, have you studied limits at all?
No... but I plan to in the near future. I hope to become a physicist when I'm a bit taller and more educated.

The average height would be the area of the circle divided by the horizontal amount of space the circle takes up, right? So (2*2*pi)/4 = pi.
Wait a minute. Can you explain that a bit more thoroughly?

@blah (right below me)
What's being taller got to do with anything?
Well... when I'm older. That was what I was saying.

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#### blah

##### brah
What's being taller got to do with anything?

#### gylve

##### Member
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

#### spdqbr

##### Member
I don't think it is mean value theorem.
Doh, I meant average value, forgetting what the mean value theorem *actually* is for a minute there. I'm blaming it on sleep deprivation from work gylve took the approach I attempted to steer Nukoca toward though, well done!

#### Lucas Garron

##### Member
This sounds like a job for calculus! Particularly the mean vale theorem.

Really, have you studied limits at all?
Mean Value Theorem? No. Have you studied the mean value theorem at all?

Anyhow, I agree with qq's derivation, which barely needs more explanation.

#### spdqbr

##### Member
Have you studied the mean value theorem at all?
Yeah, but it's been a while. Right principle wrong theorem, sorry.

#### Nukoca

##### Member
The average height would be the area of the circle divided by the horizontal amount of space the circle takes up, right? So (2*2*pi)/4 = pi.
Wait, so why would the average height be A/d?

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.

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#### TomZ

##### Member
No, you wouldn't:

A=pi*r^2
average height=pi*r^2/2r
average height squared = pi^2*r^4/4r^2

pi^2*r^4/4r^2=pi*r^2 for r=0

#### Nukoca

##### Member
average height=pi*r^2/2r
Why? Can someone please explain the logic here to me before I blow a blood vessel?

#### gylve

##### Member
I showed why the average height is area/diameter on my preview post, but you need to understand limits and riemann's integral.

#### TomZ

##### Member
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.

#### Nukoca

##### Member
Thanks for replying to me. And sorry for losing my temper there.

Before you edited your post, you said you understood qqwref. What I did was nothing other than to rephrase him in a formula form.
Well, I thought I did. But later I reread it and I realized that I didn't. 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?

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)
Kay... makes sense.

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.

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#### Stefan

##### Member
By definition dividing the area of an object by it's width will give you the height.
I'd like to see that definition, please (on let's say mathworld, wikipedia, etc).

#### TomZ

##### Member
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 made a mistake there. It should have said 2r instead of 2pi.

Now let's see if I can come up with a way to talk me out of it or vandalize wikipedia this afternoon to answer Stefan's question..

#### Nukoca

##### Member
Now let's see if I can come up with a way to talk me out of it or vandalize wikipedia this afternoon to answer Stefan's question..
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...

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