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

Correct — C, π : 1. Notice first that 18, 24 and 30 are 3, 4 and 5 each multiplied by 6, so the triangle is right-angled with the 30 cm side as hypotenuse. Two consequences follow at once. The area is half the product of the legs, that is (18 x 24)/2 = 216 square centimetres. And for a right triangle the circumcircle has the hypotenuse as its diameter, so the radius is 15 cm and the circle's area is π x 15² = 225π, about 706.9 square centimetres. The ratio of circle to triangle is 225π : 216, which is 1.0417π, about 3.27. Since π itself is about 3.14, the ratio is approximately π : 1.

• (a)1 : π — Inverts the comparison. The circle encloses the triangle, so its area must be the larger of the two; 1 : π would make the circle about a third of the triangle.
• (b)√π : 1 — The square root of π is about 1.77, so this would make the circle less than twice the triangle. The computed ratio is about 3.27.
• (d)√π : 2 — About 0.89, which would make the circle slightly smaller than the triangle it contains — impossible for a circumcircle.

Any triangle inscribed in a circle so that one side is a diameter is right-angled at the opposite vertex, and the converse holds: the circumcircle of a right triangle has the hypotenuse as its diameter. That converse is what makes this item a ten-second calculation instead of an exercise in the general formula R = abc/(4 x area).

The recognition step is the whole question. Multiples of 3-4-5 turn up constantly — 6-8-10, 9-12-15, 18-24-30, 30-40-50 — and spotting one converts an awkward triangle into a right triangle with a known area and a known circumradius. Even without that, an order-of-magnitude check disposes of three options: the circle contains the triangle, so the ratio has to exceed 1, and only π : 1 does.

• 18, 24 and 30 form a Pythagorean triple, being 3-4-5 scaled by 6.
• The circumradius of a right triangle is half its hypotenuse, so R = 15 cm here.
• Area of the triangle = (18 x 24)/2 = 216 sq cm; area of the circle = 225π ≈ 706.9 sq cm.
• The exact ratio is 225π : 216 = 1.0417π ≈ 3.27, which rounds to π : 1.
• In general R = abc/(4 x area), which gives the same 15 cm here.

• Missing that 18-24-30 is a right triangle and reaching for Heron's formula.
• Inverting the ratio and comparing triangle to circle.
• Taking the circumradius as the full hypotenuse rather than half of it.

A ratio-of-areas item that becomes trivial the moment the Pythagorean triple is recognised.

No directly related past PYQ was found.

• practice — not a real PYQ

  The circumradius of a triangle with sides 6 cm, 8 cm and 10 cm is

  • (a)4 cm
  • (b)5 cm
  • (c)6 cm
  • (d)10 cm

  Answer(b) 5 cm — it is a right triangle, so the circumradius is half the hypotenuse.

• practice — not a real PYQ

  A triangle has sides 9 cm, 12 cm and 15 cm. Its area is

  • (a)48 sq cm
  • (b)54 sq cm
  • (c)60 sq cm
  • (d)72 sq cm

  Answer(b) 54 sq cm — a 3-4-5 triangle scaled by 3, so the area is half of 9 x 12.