# Thread: Calculating segment height with only the area of segment and radius of circle

1. ## Calculating segment height with only the area of segment and radius of circle

I have been struggling to find a method to do this (that I can understand).
I have a bar with a diameter of 65mm.
It has to have a segment with a 10% cross sectional area of the full circle.
This is the data no angles or arc lengths etc.
I need to calculate the segment width or height.
Thanks in advance if any one can point me in the right direction.
I have drawn it in cad and have a rough idea it is approximately 10mm.
This was done by trial and error.
please see my jpeg image sketch for clarity.
Thankyou
Matt

2. ## Re: Calculating segment height with only the area of segment and radius of circle

$R=65$

$h$ is the value you are after

$A=R^2 \cos^{-1}\left(\dfrac{R-h}{R}\right) - (R-h)\sqrt{2 R h - h^2}$

and you want

$\dfrac{A}{\pi R^2} = 0.1$

Solving this numerically (cause it's not going to be solved any other way) Mathematica gets

$h=20.342$

3. ## Re: Calculating segment height with only the area of segment and radius of circle

Thankyou very much.
I have tried plugging the numbers in to the formula but it does not work out for me. (by the way the radius was 32.5)
My 10% area is approx 331 but when I put in R=32.5 and h=10.216 into the above formula I get a ridiculous answer.
Matt

4. ## Re: Calculating segment height with only the area of segment and radius of circle

Originally Posted by matty1971
Thankyou very much.
I have tried plugging the numbers in to the formula but it does not work out for me. (by the way the radius was 32.5)
My 10% area is approx 331 but when I put in R=32.5 and h=10.216 into the above formula I get a ridiculous answer.
Matt
looks ok to me

5. ## Re: Calculating segment height with only the area of segment and radius of circle

I forgot to change to radians.
It works on paper now.
I do not have this software.
Matt

6. ## Re: Calculating segment height with only the area of segment and radius of circle

Unfortunately my maths ability is not great and I cannot transpose the formula to solve for h.( I already know the area which is 331)
Mat

7. ## Re: Calculating segment height with only the area of segment and radius of circle

Originally Posted by matty1971
Unfortunately my maths ability is not great and I cannot transpose the formula to solve for h.( I already know the area which is 331)
Mat
that's right you can't

it's not going to have a clean closed form solution

it will have to be solved numerically each time you want to do this.