surface area of domain delimited by two lines and two non concentric circles

Question

I am looking for a way to estimate the area of this "sector-like" surface between two non concentric circles delimited by arbitrary segments:

I'm interested in both a methodology to calculate the area of the highlighted surface in Cartesian or polar coordinates and more importantly an expression or an approximation as good as possible in terms of the line segments length and arcs length.

In fact this is derived from a concrete situation where I can easily measure these parameters but only roughly estimate radius and arcs angles, and even less coordinates, needed for a more formal calculation.

Anyway, I don't even find how to start, I tried different approaches but with no success, plus I have to express this area using a limited set of accessible parameters. My math are from a long time ago, as is my english!, so any help will be appreciated.

Thank you

Edit:

Why the down vote, did you look at the picture at least to give some advice...

Anyway, I lightened the post a bit and I can try to formalize this a little more. So, consider 2 arbitrary circles and 2 arbitrary lines, how could I get the area of the domain delimited by their intersections? (shape is real, there is 4 intersection points)

• $(\mathscr{C_0}) : r=R_0$
• $(\mathscr{C_1}) : r=\delta\cos(\theta-\varphi)+\sqrt{R_1^2-\delta^2\sin^2(\theta-\varphi)}$
• $(\mathscr{D_0}) : r=d_0/\sin(\theta-\theta_0)$
• $(\mathscr{D_1}) : r=d_1/\sin(\theta-\theta_1)$

or

• $(\mathscr{C_0}) : x^2+y^2=R_0^2$
• $(\mathscr{C_1}) : (x-x_1)^2+(y-y_1)^2=R_1^2$
• $(\mathscr{D_0}) : y=a_0x+b_0$
• $(\mathscr{D_1}) : y=a_1x+b_1$

$(\delta,\varphi)=(x_1,y_1)$ position of $(\mathscr{C_1})$ from the origin
$(d_i,\theta_i)$ are distance and angle of $(\mathscr{D_i})$ from the origin
if I'm not wrong...

Answer 1

If I understand you correctly, the known information is:

• There are two circles of different radii.
• One of the circles lies inside the other, but they are not concentric.
• Two radii of the inner circle are extended to intersect the outer circle, forming an annular sector.
• the lengths of the four boundaries of that region are known.

Unfortunately this is not enough information to determine the area. Consider this thought experiment:

1. Enlarge the radius of the small circle by some small amount.
2. Leave the radius for $d_1$ where it is, but move the radius for $d_2$ around until the arc on the inner circle is back to length $L_1$.
3. On the radius for $d_1$, plot the point that is a distance of $d_1$ away from the inner circle. Move the outer circle over until it passes through this point again.
4. "Pin" the outer circle at this point. It can rotate around the point, and it can be enlarged by moving its center away from this point, but that point will continue to lie on the circle.
5. Rotate the outer circle around the pinned point until the distance between the two circles along the second radius is $d_2$.
6. If the arc on the outer circle is not $L_2$, then increase or decrease the radius by moving the center away from or towards the pinned point, until the outer circle arc is of length $L_2$
7. The previous step will have changed the other distance away from $d_2$, so go back to step 5. Keep repeating until both distances converge to the values $d_2$ and $L_2$ (and they will converge).

If you were to somehow actually do this process, you would end up with another annular sector with the same four dimensions, but a different shape. I cannot say just from this that the shape will also have a different area, but it is likely to be the case. What this demonstrates is that those four measurements are not enough to fix the shape of this region.

We need one more measurement - a measurement that is not completely determined by the four you already know. Without knowing what your situation is, I cannot say what measurements you can get. Ones that would be useful include

• the circumference of either circle.
• The straight-line distance between any two corners of your region, except the two straight sides we already know.
• straight line distances between any three well-separated points on the inner circle, or between any three well-separated points on the outer circle. (Since this introduces new points for the referents of the measurements, all three measurements will be needed.) From those distances, one can calculate the radius of one of the circles, which is enough to lock down the shape.