There are 24 equally spaced points lying on the circumference of a circle. What is the maximum number of equilateral triangles that can be drawn by taking sets of three points as the vertices?
Answer & explanation
Answer: (c) 8
An equilateral triangle inscribed in a circle has its vertices 120° apart — every 8th point of 24. Each point belongs to exactly one such triangle, so there are 24 ÷ 3 = 8.
- Adjacent points are 360° ÷ 24 = 15° apart.
- An inscribed equilateral triangle needs vertices 120° apart: 120 ÷ 15 = 8 steps.
- Starting at points 1 to 8 gives different triangles (1-9-17, 2-10-18, …, 8-16-24); starting at 9 repeats the first.
- Number of triangles = 24 ÷ 3 = 8.
Remember · On n equally spaced points, inscribed regular k-gons exist only if k divides n, and there are n/k of them.
Question and answer: UPSC's official GS Paper II (2018, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·