1
$\begingroup$

I have two circles, one of which is completely within the other. They do not touch, but are not necessarily concentric. I am given the sum of their circumferences, and the difference in their areas (ie, the area of the space inside the outer circle and outside the inner circle). I need to find the average distance between the circles or, equivalently, the radii of the circles.

In other words, given $C = C_o + C_i$, and $A = A_o - A_i$, find $R_o$ and $R_i$.

I've gotten as far as expressing $A$ in terms of $C_o$ and $C_i$, but then my long-unused algebra fails me. Any help? Thanks...

  • 1
    What *is* the average distance between two non-concentric shapes, anyway?2012-07-31

1 Answers 1

3

Recall the area is $\pi r^2$ and circumference is $2\pi r.$ The sum of circumferences gives you equation: $ r_1 + r_2 = \frac{C}{2\pi}. \tag{1}$ The difference of areas gives a second equation: $ r_1^2 - r_2^2 = \frac{A}{\pi}. \tag{2}$ Factor LHS of $(2)$ into $(r_1 - r_2)(r_1 + r_2),$ substitute the value of $r_1 + r_2$ from $(1),$ and you will then have 2 linear equations in two unknowns, which you can solve.

  • 0
    It's a shame; I was really good at this stuff in high school. You really do lose it if you don't use it. Thanks so much for the help.2012-07-31