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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 2023 · Q67

Hard

A rectangular floor measures 4 m in length and 2.2 m in breadth. Tiles of size 140 cm by 60 cm have to be laid such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. What is the maximum number of tiles that can be accommodated on the floor?

Answer & explanation

Answer: (d) 9

Any line across the 220 cm breadth meets tile sides of 60 cm or 140 cm, which can cover at most 200 cm of it. So at most 400 × 200 = 80,000 cm² can be tiled, room for fewer than 10 tiles of 8,400 cm², and a staggered layout fits 9.

  1. Floor = 400 cm × 220 cm; one tile = 140 cm × 60 cm = 8,400 cm².
  2. Across the 220 cm breadth, tile sides add up in 60s and 140s: the best totals are 60 + 140 = 200 and 60 + 60 + 60 = 180. So every line across the breadth leaves at least 20 cm uncovered.
  3. Covered area ≤ 400 × 200 = 80,000 cm², so tiles ≤ 80,000 ÷ 8,400 ≈ 9.5, i.e. at most 9.
  4. A layout of 9 (length measured 0–400, breadth 0–220): three tiles lying lengthwise, stacked, at length 0–140 (breadth 0–180); one lengthwise tile at length 140–280, breadth 0–60; two standing tiles (60 along the length) at length 140–260, breadth 60–200; two standing tiles at length 280–400, breadth 0–140; one lengthwise tile at length 260–400, breadth 140–200.
  5. 3 + 1 + 2 + 2 + 1 = 9 tiles, none overlapping — the maximum.

Remember · For tiling maxima, check a 'line' bound as well as area: the lengths that fit across a side often beat the area estimate.

Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

CSAT 2023 · Q79

Medium

A cuboid of dimensions 7 cm × 5 cm × 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm × 5 cm, 5 cm × 3 cm, 7 cm × 3 cm respectively. Then the cuboid is cut and separated into various cubes each of side length 1 cm. Which of the following statements is/are correct?

  1. 1.There are exactly 15 small cubes with no paint on any face.
  2. 2.There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.

Select the correct answer using the code given below:

Answer & explanation

Answer: (a) 1 only

The unpainted cubes form an inner block of 5 × 3 × 1 = 15. Green (5 × 3) and blue (7 × 3) faces meet only along the four edges of length 3 cm, and each such edge gives 3 − 2 = 1 cube with exactly those two faces painted — 4 cubes, not 6.

  1. No paint: remove one layer from each side: (7 − 2) × (5 − 2) × (3 − 2) = 5 × 3 × 1 = 15. Statement 1 is correct.
  2. Green faces are the 5 × 3 ends (across the 7 cm length); blue faces are the 7 × 3 sides (across the 5 cm width).
  3. A green face meets a blue face along an edge 3 cm long; a cuboid has 4 such edges.
  4. Each 3 cm edge has 3 cubes; the 2 end ones are corners with three painted faces, leaving 1 cube per edge.
  5. Blue-and-green-only cubes = 4 × 1 = 4, not 6. Statement 2 is incorrect.
  6. Check: two-face cubes in all = 4(7 − 2) + 4(5 − 2) + 4(3 − 2) = 20 + 12 + 4 = 36, and the green–blue share is the 4 on the 3 cm edges.
  • ✓ 1. Inner block (7 − 2)(5 − 2)(3 − 2) = 15 cubes.
  • ✗ 2. Green–blue edges are the four 3 cm edges, each giving one such cube: 4 in all.

Remember · Painted cuboid a × b × c: unpainted = (a − 2)(b − 2)(c − 2); cubes on an edge with two colours = edge length − 2.

Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

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