Chord Length of Intersecting Circles

chdlen.html

Exploration

Graphics saved as "cir19.eps".

Show that the distance, d, between the intersection points of two circles is given by

where D is the distance between the centers of the circles, and and are the radii of the two circles.

Approach

Assume the radii of the two circles centered at and are and , respectively, is one of the intersection points, and the distance between the centers is D. The length of the common chord, d, can be found by equating the area (squared) of using Heron's formula and the standard area formula A=b h/2.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

is the area (squared) by Heron's formula.

Clear[r1,r2,D1];
s=(r1+r2+D1)/2;
A1=s(s-r1)(s-r2)(s-D1) //Simplify

is the area (squared) by the standard area formula A=b h/2 (d is the distance between the intersection points, i.e. the length of the chord).

Clear[d];
A2=(D1*(d/2)/2)^2 //Simplify

Set the areas equal and solve for d. Take the positive value. Simplify the result involving with an algebraically equivalent expression.

ans=Solve[A1==A2,d]

Discussion

If the radii are equal the result can be significantly simplified.

Clear[r];
ans2=ans1 //. {r1->r, r2->r}