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CSAT · 20 questions

Geometry & mensuration

Every UPSC CSAT question on this topic, 2016–2026, newest first. Tap an option to check yourself; the answer and explanation open below it.

Geometry & mensuration questions per year: 2016: 6, 2017: 1, 2018: 3, 2019: 0, 2020: 3, 2021: 2, 2022: 2, 2023: 2, 2024: 1, 2025: 0, 2026: 0 Asked in 8 of 11 years · most in 2016 (6)

UPSC syllabus: “Basic numeracy (numbers and their relations, orders of magnitude, etc.) (Class X level), Data interpretation (charts, graphs, tables, data sufficiency etc. — Class X level);” See the full syllabus →

CSAT 2018 · Q35

Easy

A solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side. How many of the small cubes will have exactly two painted faces?

Answer & explanation

Answer: (a) 12

Small cubes with exactly two painted faces sit along the edges of the big cube, excluding the corners. A cube has 12 edges, each with 3 − 2 = 1 such cube here, giving 12.

  1. The cube splits into 3 × 3 × 3 = 27 small cubes.
  2. Corner cubes have 3 painted faces; edge cubes other than corners have exactly 2.
  3. Each edge has 3 small cubes, 2 of them corners, leaving 1 per edge.
  4. 12 edges × 1 = 12 cubes with exactly two painted faces.
  5. Check: 8 (three faces) + 12 (two) + 6 (one) + 1 (none) = 27.

Remember · Painted n-cube: 8 corners with three faces, 12(n − 2) edge cubes with two, 6(n − 2)² with one.

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). Permalink ·

CSAT 2018 · Q58

Easy

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.

  1. Adjacent points are 360° ÷ 24 = 15° apart.
  2. An inscribed equilateral triangle needs vertices 120° apart: 120 ÷ 15 = 8 steps.
  3. Starting at points 1 to 8 gives different triangles (1-9-17, 2-10-18, …, 8-16-24); starting at 9 repeats the first.
  4. 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). Permalink ·

CSAT 2018 · Q60

Easy

Twelve equal squares are placed to fit in a rectangle of diagonal 5 cm. There are three rows containing four squares each. No gaps are left between adjacent squares. What is the area of each square?

Answer & explanation

Answer: (c) 1 sq cm

With squares of side s, the rectangle measures 4s by 3s, so its diagonal is 5s by Pythagoras. 5s = 5 cm gives s = 1 cm and an area of 1 sq cm.

  1. Let each square have side s; the rectangle is 4s long and 3s wide.
  2. Diagonal = √((4s)² + (3s)²) = 5s.
  3. 5s = 5, so s = 1 cm and each square's area is 1 sq cm.
  4. Check: a 4 cm × 3 cm rectangle has diagonal √25 = 5 cm.

Remember · Spot the 3-4-5 triangle: sides in the ratio 3 : 4 make the diagonal 5 parts.

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). Permalink ·

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