Multiple choice

The area of a circle inscribed in an equilateral triangle is $154$ $cm^2$. Find the perimeter of the triangle.

  1. $46.3$ cm
  2. $66.3$ cm
  3. $72.7$ cm
  4. $78.8$ cm
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C Correct answer
Explanation

Area of circle = pi*r^2 = 154. r^2 = 154 / (22/7) = 49. r = 7. For an equilateral triangle, the inradius r = side / (2*sqrt(3)). 7 = side / (2*sqrt(3)). Side = 14*sqrt(3) = 14 * 1.732 = 24.248. Perimeter = 3 * side = 3 * 24.248 = 72.744 cm.

AI explanation

The area of the incircle is 154, so using the area of a circle formula we have pi times r squared equals 154, meaning r squared is 49 and r is 7. For an equilateral triangle, the relationship between the inradius and the side length is r equals s divided by the square root of 12. Substituting r equals 7 gives a side length s of 7 times the square root of 12, which is 14 times the square root of 3. The perimeter is 3 times s, resulting in 42 times the square root of 3, which is approximately 72.7 cm.

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