Find the diameter (in cm) of a circle whose circumference is equal to the sum of the circumferences of two circles of radii 12 cm and 16 cm.

Correct — D. Circumference is 2πr, a linear measure, so π cancels straight out.

2πR = 2π(12) + 2π(16)
R = 12 + 16 = 28 cm
Diameter = 2R = 56 cm → option (d)

One step shorter: since C = πd, adding circumferences adds diameters, and 24 + 32 = 56.

Why the others are wrong

• (a)
  28 — 28 — that is R, the radius you get from 12 + 16. The question asks for the diameter, which is twice it, so this is the answer stopped one line early.
• (b)
  32 — 32 — the diameter of the 16 cm circle on its own, 2 × 16. The 12 cm circle has been dropped out of the sum altogether.
• (c)
  64 — 64 — larger than the two given diameters combined, 24 + 32 = 56. The new circumference is fixed at 56π, so its diameter can only be 56.

Circumference is directly proportional to radius: C = 2πr = πd. Adding circumferences therefore adds radii and adds diameters, with no π and no squaring anywhere in the working.

Area behaves differently. It goes as r², so a circle whose area equals the sum of two areas has R = √(12² + 16²) = 20 cm, not 28 — a different question with a different answer.

Deciding which of the two you are in settles the problem before any arithmetic starts.

Key facts

• C = 2πr = πd, so circumference is proportional to the radius.
• A circle whose circumference is the sum of two others has radius r1 + r2.
• For areas the rule differs: R = √(r1² + r2²), which for 12 cm and 16 cm gives 20 cm.

Common traps

• Answering 28, the radius, when the diameter was asked for
• Squaring the radii out of habit, because most circle questions are about area
• Cancelling 2π on one side only and losing a factor of 2

No directly related past PYQ was found.