Multiple choice

The length of side of an equilateral triangle is $12$ cm. Find the ratio of the areas of the circle circumscribing the triangle to the circle inscribing the triangle

  1. $2:1$
  2. $5:4$
  3. $4:1$
  4. $3:2$
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C Correct answer
Explanation

For an equilateral triangle, the radius of the circumcircle R = a/sqrt(3) and the radius of the incircle r = a/(2*sqrt(3)). The ratio of areas is (pi*R^2) / (pi*r^2) = (R/r)^2 = (2)^2 = 4:1.

AI explanation

The radius of the circumcircle (R) of an equilateral triangle is side / sqrt(3), and the radius of the incircle (r) is side / (2 * sqrt(3)). For a side length of 12, R = 12 / sqrt(3) and r = 6 / sqrt(3). The ratio of their areas is R^2 / r^2, which equals (12 / sqrt(3))^2 / (6 / sqrt(3))^2 = 144 / 36 = 4. The ratio of the areas of the circumscribing circle to the inscribing circle is 4:1.

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