Minimalist IAS
CSAT 2018 paper

UPSC CSE CSAT 2018 · Question 1 · Puzzles, arrangements & general mental ability

Consider the following three-dimensional figure:

CSAT 2018 · Q1

Puzzles, arrangements & general mental ability Medium

Consider the following three-dimensional figure:

A regular icosahedron drawn as a wire frame, with hidden edges dashed: 12 vertices, 30 edges and triangular faces all round.
From UPSC's question paper.

How many triangles does the above figure have?

Answer & explanation

Answer: (b) 20

The solid is a regular icosahedron, and every one of its faces is a triangle. Its edges close into no triangles other than the faces, so the count is simply the number of faces: 20.

  1. Identify the solid: every face is a triangle and five faces meet at each vertex — a regular icosahedron.
  2. Count the faces in three layers: 5 around the top vertex, 5 around the bottom vertex and a band of 10 in the middle.
  3. 5 + 10 + 5 = 20 triangular faces.
  4. No three edges outside a face close into a triangle, so there are no extra triangles to add.
  5. Check: Euler's formula V − E + F = 2 with 12 vertices and 30 edges gives F = 20.

Remember · For a solid made of triangular faces, count faces layer by layer and confirm with Euler's formula V − E + F = 2.

Question and answer: UPSC's official GS Paper II (2018, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 1 Oct 2026 (how we verify). ·

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