Circumscriptive Geometry

About a unit circle, a regular hexagon is circumscribed, about which another circle is circumscribed, about which an equilateral triangle is circumscribed, about which a third circle is circumscribed. Find, in simplified and exact form, the ratio of the area of the smallest circle to that of the largest.

Note: within a regular polygon, the distance from its center to any given vertex is called its radius. The length of a perpendicular from its center to any given side is called its apothem.

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Circumscriptive Geometry Solution

Bear in mind that the radius of a regular polygon is the same as the radius of its circumscribed circle. The radius of the hexagon can be determined by locating a 30-60-90 triangle; its longer leg is the apothem, 1, its shorter leg is half the length of a side of the hexagon, and its hypotenuse is the radius. The ratio of the shorter leg to the longer is 1:√3, so the shorter leg is

√3

, and the hypotenuse, being twice the shorter leg, is

√3

. The apothem of the equilateral triangle, then, is also

√3

, and by again locating a 30-60-90 triangle, with which the apothem is the shorter leg and half the length of a side of the triangle is the longer leg, we find the hypotenuse and radius of both the triangle and its circumscribed circle to be twice

√3

. Since the ratio of the circles' radii is √3:4, the ratio of their areas is 3:16.

Answer: 3:16

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