Multiple choice

A circle is inscribed in an equilateral triangle of side 12 cm. A square is then inscribed inside this circle such that all vertices of the square touch the circle. What is the area of the square?

    1. 16 cm2

  2. 18 cm2

  3. 24 cm2

  4. 36 cm2
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C Correct answer
Explanation

The radius of the inscribed circle of an equilateral triangle with side s is r = s / (2*sqrt(3)). For s=12, r = 12 / (2*sqrt(3)) = 2*sqrt(3). A square inscribed in a circle of radius r has a diagonal d = 2r = 4*sqrt(3). The area of the square is d^2 / 2 = (4*sqrt(3))^2 / 2 = (16*3) / 2 = 24.

AI explanation

The radius of the inscribed circle in an equilateral triangle with side length s is s times the square root of 3, all divided by 6. For a 12 cm side, the radius is 2 times the square root of 3 cm. The diagonal of the square inscribed in this circle equals the circle's diameter, which is 4 times the square root of 3 cm. The area of the square is half of its diagonal squared, so 0.5 times (4 times the square root of 3)^2, which equals 24 square centimeters. The result is 24 square centimeters.

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