Geometry - circumference of a ring with a cross?

Question

Hello,

I am trying to calculate the circumference of a ring with a cross cut out. For better understanding see this drawing:
http://img14.imgspot.com/u/07/234/16/slottedtube.gif

Known dimensions are D, d1 and d2. I want to calculate the total length of lines and curves marked in RED color.

Based on the knowledge gained from one of my previously asked question I managed to come up with a foruma to calculate this. I would appreciate if you could have a look and tell me if my formula is correct or did I make a mistake?

Here goes:

The total length L is:
L = 8 * X + 4 * Y

X = 1/2 * SQRT(D^2 - d2^2) - 1/2 * SQRT(d1^2 - d2^2)

Y = (90 - 2 * arcsin(d2/d1) ) * PI * d1 / 360

Is this correct? If someone can calculate the surface on his/her own and let me know if my approach is correct that would be great.

Thanks in advance!

The D is diameter of the circle, my mistake in the drawing.

For Y in radians it should be 2*pi/4 not just pi/4, right?

Answer 1

Your expression for Y is quite correct (provided you take the value of arc sin in degrees what is not the usual agreement) or, even better
Y =(pi/4 - arc sin(d2/d1))*d1 in radians.
But the expression you've supplied for X would be correct if D is the length of the DIAMETER of the larger circle. Looking at the figure I could not be sure whether that's so, or D is the distance between the endpoints of the cross - according the drawing I'd rather suspect the latter, then the expression for X would be
X = D/2 - (1/2)*SQRT(d1^2 - d2^2).

To avoid misunderstandings if D was intended to be diameter length, I'd picture it neither horizontally, nor vertically, but in "Ø"-shape like d1 is drawn instead.