A triangle of sides 18 cm, 24 cm and 30 cm is inscribed in a circle. The ratio of the area of the circle to that of the triangle is approximately

The triangle has sides 18 cm, 24 cm, and 30 cm. Since 18² + 24² = 324 + 576 = 900 = 30², it is a right-angled triangle. For a right triangle, the circumradius (R) is half the length of the hypotenuse. Thus, R = 30 / 2 = 15 cm. The area of the circumscribed circle is πR² = π(15)² = 225π cm². The area of the right triangle is (1/2) × base × height = (1/2) × 18 × 24 = 216 cm² [1]. The ratio of the circle's area to the triangle's area is 225π / 216. Simplifying the fraction 225/216 by dividing both by 9 gives 25/24. Therefore, the ratio is (25/24)π, which is approximately 1.04π. Among the given options, π:1 (approximately 3.14) is the closest approximation to 3.27.

Sources

1. [1] https://r-knott.surrey.ac.uk/pythag/pythag.html