Multiple choice

If $p$ is the perimeter of an equilateral triangle inscribed in a circle, the area of the circle is:

  1. $\cfrac { \pi { p }^{ 2 } }{ 3 } $
  2. $\cfrac { \pi { p }^{ 2 } }{ 9 } $
  3. $\cfrac { \pi { p }^{ 2 } }{ 27 } $
  4. $\cfrac { \pi { p }^{ 2 } }{ 81 } $
  5. $\cfrac { \pi { p }^{ 2 }\sqrt { 3 } }{ 27 } $
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C Correct answer
Explanation

For an equilateral triangle with side s, perimeter p = 3s, so s = p/3. The radius of the circumcircle R = s / sqrt(3) = p / (3 * sqrt(3)). Area of circle = pi * R^2 = pi * (p^2 / (9 * 3)) = pi * p^2 / 27.

AI explanation

Each side of the equilateral triangle is p divided by 3. The radius of the circumscribed circle (R) relates to the side (a) by the formula R = a divided by the square root of 3, so R equals p divided by (3 times the square root of 3). Substituting this into the circle area formula gives pi times R squared, resulting in (pi times p squared) divided by 27.

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