A yellow circle
Geometry
Level
2
A blue equilateral triangle is inscribed in a red circle. Then a yellow circle is inscribed in this blue triangle. If the radius of the red circle is one unit, what is the area of the yellow circle in square units?
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1 solution
Draw a right triangle with one vertex as the center of the red circle (and the yellow circle), one vertex touching a vertex of the triangle, and the final vertex touching the point of tangency between the circle and the triangle. You form a 30-60-90 triangle. Since the hypotenuse is 1, the 60-degree side, which is the radius of the small circle, is 1/2. The formula for the area of a circle is πr^2, and r=1/2, π(1/2)^2 = π/4. As a result, the answer is π/4.