Problem of the Week: Bull's-eye
The Locus Archery Supply Company manufactures archery targets in several different shapes. However, every target they produce has a common property: the boundary of the bull's-eye is the set of points equidistant from the center of the target and the outside perimeter of the target. This means that the bull's-eye on one of their circular targets is a circle whose radius is half the radius of the target; the area of the bull's-eye is 1/4 the area of the whole target. (See the picture below; the bull's-eye is the red area.)
The company does not limit itself to circular targets, however. They also produce a target in the shape of an equilateral triangle, with the same criterion for the boundary of the bull's-eye. For one of these triangular targets, what is the ratio of the area of the bull's-eye to the area of the target?
(For an extra challenge, find a formula for the ratio of the area of the bull's-eye to the area of the target when the target is in the shape of a regular n-gon.)
Winner
The winning solution to this problem was by Rob Brase.
