SOLUTION: A circle is inscribed in a square, which is inscribed in an equilateral triangle inscribed in another circle. What is the ratio of the area of the inner circle to the area of the o

Question 1190325: A circle is inscribed in a square, which is inscribed in an equilateral triangle inscribed in another circle. What is the ratio of the area of the inner circle to the area of the outer circle?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Let the equilateral triangle be ABC with base AC and peak B. Let BD be the altitude of the triangle to side AC. Let EF be the side of the square parallel to base AC but not on AC; EF intersects BD at G.

Let r = OG be the radius of the inner circle.

Triangle BGF is a 30-60-90 right triangle; and the length of GF is the radius of the circle, r; so .

BO is the radius of the outer circle; and

The radii of the inner and outer circle are then and , so the ratio of the areas of the inner and outer circles is

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