SOLUTION: What is the ratio of the area of an equilateral triangle to that of a regular hexagon inscribed in the same circle?

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the radius of the circle is 2/3 of the distance from the vertex of the triangle to the opposite side (height)

so the height of the triangle is 1.5r __ area=(1/2)((r)sqrt(3))(1.5r)

the hexagon is made up of six equilateral triangles whose sides are equal to the radius of the circle
__ area=6(1/2)(r)((r/2)sqrt(3))