About thickness: adding circumferences to account for bangs? How does that work??

[Magalo]

Hi guys!!

I really suck at math so I'm not sure how to do this.

Ponytail circumference is exactly 3", but I have very thick (Bettie Paige style) bangs and I decided to measure them for fun a couple of minutes ago. It ended up at 1 1/4".

Is my total circumference 4 1/4? It is that simple? I feel like it's the incorrect way to do this! :p

[LAG93]

No I think it is more complicated than that, you would have to add each area together and then with the new area find the true circumference. It's possible but with hair maybe quite difficult to actually measure!

[Firefox7275]

The most accurate way is to guesstimate, if your bangs were to join your ponytail they would have thinned out anyway.

[Audhumla]

I know I know.
Ok, so adding the circumferences is wrong.
This is because the circumference it proportional to the diameter and you can see (from the picture below) that if you just added the diameters you'd get a circle (with a circumference equal to the sum of the circumferences of the smaller circles) that could "fit" a lot more hair than the smaller circles put together.
http://media-cache-ak0.pinimg.com/736x/d3/a1/d6/d3a1d6fc911530baaadfe1ee4f5145d8.jpg
So what you need to do is add the ponytails in a way that you get a ponytail with a cross sectional area that is equal to the sum of the cross sections of the ponytails you're adding.
In your case you have 3" circumference and 1.25" circumference.
The area of a circle is given by pi x the radius squared and the circumference is pi times the diameter. So we need to backtrack from the circumferences to get a radius for both the ponytails and then figure out the area of the cross section of the two ponytails and then add that area and then go in reverse to find the circumference for a circle with that new area.
So for 3" (I'm a metric gal but I'll work this in inches for simplicity) we want to divide by pi to get the diameter. This gives ~0.955"
Then half that to get the radius. ~0.477
Then we use that radius, square it a multiply by pi to get the area. This gives ~0.716"^2
Use the same method for the 1.25" pony. Add the areas to get a total area of ~0.841"^2
Working the other way from that area, divide it by pi then take the square root to give a radius of ~0.517".
Double to find the diameter then multiply by pi to get the new circumference.
The answer is *drumroll* 3.25"
Sorry it's not 4.25" :p

Wow did I do that the long way or what.
But it helps you see how you get the answer I hope.

[Beborani]

Sq rt of 3 square plus 1.25 square which equals 3.25 assuming all your bangs get to your ponytail

[0xalis]

I was wondering the same thing myself! Thanks, math geniuses! :p

[biogirl87]

emilylightning, an easier way to do this would be to divide the circumference of each ponytail by 2 pi to get the radius of each ponytail, then add the radii (plural of radius, I believe) of the ponytailis together, and then multiply the total radius by 2 pi to get the circumference of how big the ponytail would be if you were to combine bangs and the main part of the hair. I think figuring out the area and then dividing the area by pi and taking the square root of the result adds an extra step that I don't think is necessary in this calculation. I could be wrong, of course, but I doubt it.

[Audhumla]

emilylightning, an easier way to do this would be to divide the circumference of each ponytail by 2 pi to get the radius of each ponytail, then add the radii (plural of radius, I believe) of the ponytailis together, and then multiply the total radius by 2 pi to get the circumference of how big the ponytail would be if you were to combine bangs and the main part of the hair. I think figuring out the area and then dividing the area by pi and taking the square root of the result adds an extra step that I don't think is necessary in this calculation. I could be wrong, of course, but I doubt it.
I know the way I did it isn't the most efficient way in the world but adding the radii is the same as adding the circumferences because they're proportional right?
Adding the radii and then finding the circumference of that circle gives 4.25" as well.

[biogirl87]

I know the way I did it isn't the most efficient way in the world but adding the radii is the same as adding the circumferences because they're proportional right?
Adding the radii and then finding the circumference of that circle gives 4.25" as well.You seem to be to be right, but I just redid the calculation and I don't understand why I'm coming up with 4.25". I don't want to get this thread off topic, so feel free to PM me and we can discuss this further.

[biogirl87]

I know the way I did it isn't the most efficient way in the world but adding the radii is the same as adding the circumferences because they're proportional right?
Adding the radii and then finding the circumference of that circle gives 4.25" as well.You seem to be to be right, but I just redid the calculation and I don't understand why I'm coming up with 4.25". I don't want to get this thread off topic, so feel free to PM me and we can discuss this further.

[Audhumla]

You seem to be to be right, but I just redid the calculation and I don't understand why I'm coming up with 4.25". I don't want to get this thread off topic, so feel free to PM me and we can discuss this further.
I think it's ok to reply here since this is what the thread's about.
So yeah I'm not advocating my method. The long way around is usually the one that occurs to me first for whatever reason.
However, we can all agree that adding the circumferences is wrong. Now the circumference is π x the diameter or 2 x π x the radius.
(Using R for larger radius and r for smaller radius. π is pi in this font.) It's the same thing to add the circumferences (2πR + 2πr) which you could simplify to 2π(R+r) or add the radii and then multiply to get the circumference of that circle (r+R) x 2π.
The way Beborani worked it is correct and a lot shorter. Taking the square root of the sum of the squares of the circumferences.
The difference between the right way and the wrong way is the difference between adding two numbers or squaring two numbers and then taking the square root of the sum of the squares.
eg. 3+5 = 8 but √(3²+5²) = ~6
Following the explanation that I gave earlier in the thread you have to consider an approach that is equivalent to adding the areas and not the circumferences and the area is proportional to r² not r (Area of a circle= πr²) and because you can just multiply the radius by 2π which is a scalar quantity(not involving a variable) to get the circumference then the area must be proportional the the circumference squared too and not just the circumference.
Feel free to correct me anyone but I'm pretty sure that's right. Makes sense to me.

[biogirl87]

I think it's ok to reply here since this is what the thread's about.
So yeah I'm not advocating my method. The long way around is usually the one that occurs to me first for whatever reason.
However, we can all agree that adding the circumferences is wrong. Now the circumference is π x the diameter or 2 x π x the radius.
(Using R for larger radius and r for smaller radius. π is pi in this font.) It's the same thing to add the circumferences (2πR + 2πr) which you could simplify to 2π(R+r) or add the radii and then multiply to get the circumference of that circle (r+R) x 2π.
The way Beborani worked it is correct and a lot shorter. Taking the square root of the sum of the squares of the circumferences.That may be it (it's late at night and I think my brain doesn't work well at this hour), but for whatever reason when I calculated the circumference of my total ponytail (when my bangs still didn't fit in the ponytail), from the individual circumferences I calculated the individual radii, added the individual radii together and used the sum of the two radii for the calculation of the total circumference and seemed to come up with the correct circumference. I might measure my ponytail circumferences (main part of hair and bangs) and post them on this thread (if the original poster doesn't mind or in another thread and have you ladies help me out with the calculation).

[Audhumla]

That may be it (it's late at night and I think my brain doesn't work well at this hour), but for whatever reason when I calculated the circumference of my total ponytail (when my bangs still didn't fit in the ponytail), from the individual circumferences I calculated the individual radii, added the individual radii together and used the sum of the two radii for the calculation of the total circumference and seemed to come up with the correct circumference. I might measure my ponytail circumferences (main part of hair and bangs) and post them on this thread (if the original poster doesn't mind or in another thread and have you ladies help me out with the calculation).
Unless OP objects I volunteer to provide such calculations to anyone who wants them done :p
If you don't want to clutter up the thread feel free to PM me :D

[Magalo]

I know I know.
Ok, so adding the circumferences is wrong.
This is because the circumference it proportional to the diameter and you can see (from the picture below) that if you just added the diameters you'd get a circle (with a circumference equal to the sum of the circumferences of the smaller circles) that could "fit" a lot more hair than the smaller circles put together.
http://media-cache-ak0.pinimg.com/736x/d3/a1/d6/d3a1d6fc911530baaadfe1ee4f5145d8.jpg
So what you need to do is add the ponytails in a way that you get a ponytail with a cross sectional area that is equal to the sum of the cross sections of the ponytails you're adding.
In your case you have 3" circumference and 1.25" circumference.
The area of a circle is given by pi x the radius squared and the circumference is pi times the diameter. So we need to backtrack from the circumferences to get a radius for both the ponytails and then figure out the area of the cross section of the two ponytails and then add that area and then go in reverse to find the circumference for a circle with that new area.
So for 3" (I'm a metric gal but I'll work this in inches for simplicity) we want to divide by pi to get the diameter. This gives ~0.955"
Then half that to get the radius. ~0.477
Then we use that radius, square it a multiply by pi to get the area. This gives ~0.716"^2
Use the same method for the 1.25" pony. Add the areas to get a total area of ~0.841"^2
Working the other way from that area, divide it by pi then take the square root to give a radius of ~0.517".
Double to find the diameter then multiply by pi to get the new circumference.
The answer is *drumroll* 3.25"
Sorry it's not 4.25" :p

Wow did I do that the long way or what.
But it helps you see how you get the answer I hope.


Thank you so much!!
I slept on it and thought about a similar formula (finding the radius...) but really, when I graduated from high school I just erased all mathematics from my memory!! Haha!

And if some people want their total circumference calculated I don't mind if they use my thread! :p I have my answer!