## 13 August 2012

### Geometry puzzle

Futility Closet challenges you to calculate the area of the ring (the area between the two concentric circles).  The only information you have is that the green line is 42 feet long.  Answer at the link.

1. Whoa, I can't believe I got that right, except I used calculus rather than trigonometry.

That is, if the inner circle were to become infinitely small, the outer circle will become closer to having its diameter exactly that of the 42 foot green line. No matter the size of the two circles, the area of the ring will always by the same. So I set the diameter of the inner circle to approach 0 and the outer circle's diameter to approach 42 feet, and then take the area of the outer circle.

It would've ended up as 21²pi - 0 = 21²pi.

2. Assuming this is solvable with only one the length of the (presumably) straight line, the inside hollow would need to be scalable and still provide the same area answer. So if you scale the hollow down to zero diameter, the line is then the diameter of the circle. So the area would be Pi time the radius(21')squared or roughly 1384 square feet. I have no idea how to go about proving that mathematically.

3. I still don't know how big the area is. What size are his feet?

4. I used geometry to solve, instead of calculus. Draw a right triangle whose sides are (1) the line from the center point of the inner circle to the tangent point of the line to the inner circle (2) half of the green line and (3) the hypotenuse that connects those two sides. By the pythagorean theorem, that hypotenuse squared, H^2 = R^2 + 21^2, where R is the radius of the inner circle and H is the radius of the outer circle.

The area of the ring is A = pi ( H^2 - R^2).
Substituting, we get A = pi ( R^2 + 21^2 - R^2) = pi (21^2)

5. Its very simple
just drop a perpendicular from the center of inner circle to the line (at midpoint).
now connect the hypotenuse.
by Pythagorean theorem h^2=b^2 + p^2
R^2 = r^2 + 21^2
R^2 - r^2 = 441
now multiply pi to both side
so equation will become area of big circle - area of inner circle = pi * 441 =1386.