Kissing Circles For Valentines Day

The combined area of the circles is $2πR^2$ , and the formula for the area of a triangle is $\frac{1}{2}bh$

The base of the triangle is 2R (because line AB is tangent to the circles) and the height is R, so the area of the triangle becomes $\frac{1}{2} \times 2R \times R=R^2$

Therefore $cR^2=2πR^2$ so c is 2π, or $\boxed{6.2832}$

It makes sense that AC and CB would be equal to $\sqrt {2}r$ , but we can confirm that easily be constructing a perpendicular on points B, A, and C. You will have a rectangle with right angles, proving tha AC and CB are right triangles, and that AB=2r. So to find the final area of the triangle, we take $\frac {(\sqrt{2}r)^{2}}{2}=r^{2}$ . The area of the two circles will of course be $2\pi r^{2}$ , so we find the final answer: $2 \pi$ .