Area of intersection between 4 overlapping circles.

Question

I'm having difficulties finding the are of a section on the 4th circle when 4 circles intersect. The circles have a diameter of 150 mm, and the centers of adjacent circles are 100 mm apart. The shaded area is what I'm interested in. Thank you!

Answer 1

Finding the intersection points between any two of the circles is not a big deal, by analytical geometry or trigonometry. Nor is finding the delimiting angles of the arcs around their respective centers.

Let two circles $(x_c, y_c, r)$ and $(x'_c, y'_c, r')$. Plug the parametric equation of the first into the implicit equation of the second:

$$(x_c+r\cos\theta-x'_c)^2+(y_c+r\sin\theta-y'_c)^2={r'}^2,$$ or after rearranging,$$2r\Delta_x\cos\theta+2r\Delta_y\sin\theta={r'}^2-r^2-\Delta_x^2-\Delta_y^2.$$

This trigonometric equation has the form $a\cos\theta+b\sin\theta=c$ and is solved by transforming it to$$\cos(\theta-\arctan\frac ba)=\frac c{\sqrt{a^2+b^2}}.$$

You will find the shaded area by using Green's theorem for areas: http://en.wikipedia.org/wiki/Green%27s_theorem#Area_Calculation.

For a single arc, the contribution to the integral is $$\frac12\oint_{Arc} x\ dy-y\ dx=\frac r2\int_{\theta_0}^{\theta_1} (x_c\cos\theta+y_c\sin\theta+r)\ d\theta=\frac r2\ (x_c\sin\theta-y_c\cos\theta+r\theta)\vert_{\theta_0}^{\theta_1}.$$