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Geometry & Mensuration — UPSC CSAT Paper II Quantitative Aptitude previous year questions
20 free solved previous year questions on Geometry & Mensuration, papers from 2016 to 2026, drawn from UPSC CSAT, covering questions of the kind UPSC asks in CSAT Paper II. Every answer key has been checked again against published sources and solved independently, and each explanation is written fresh.
Read all 20 questions with answers
- #1UPSC CSAT 2026A cut on a solid object divides the object into two parts where the new surfaces thus produced are plane. On the other hand, one single cut can be used to cut more than one object at a time. In an experiment, the total number of pieces produced by applying n cuts is denoted by xₙ. The experiment is performed on a solid cube where pieces remain unmoved after each cut. In this experiment, if after the third cut, the pieces are identical, then which of the following is not a possible value for x₄?
- A16
- B12
- C8
- D5
Answer: (A) 16Explanation
Three cuts that give identical pieces make 4 slabs, 6 bricks or 8 small cubes.
From 4 slabs, the fourth cut can cut 1 to 4 slabs, giving 5 to 8 pieces. So 5 and 8 are possible.
From 8 small cubes (2 × 2 × 2), a plane can cut at most 6 cubes (a plane through the centre).
So the most pieces we can get is 8 + 6 = 14.
12 is possible by cutting 4 cubes.
16 would need 8 more pieces, but a plane cuts at most 6 cubes. So 16 is not possible.
Answer: 16, option (a).Source: UPSC CSAT 2026 Paper II (Set A); answer key checked by independent computation and against UPSC's provisional key
- #2UPSC CSAT 2023125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes from each side?
- A27
- B25
- C21
- D18
Answer: (A) 27Explanation
125 cubes form a 5 x 5 x 5 block.
A cube is surrounded on all sides only if it is not on the outer layer.
The inner block is (5-2)^3 = 27 cubes.
Answer: option (a).Source: UPSC CSAT 2023 question; answer key checked by independent computation
- #3UPSC CSAT 2023A cuboid of dimensions 7 cm x 5 cm x 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm x 5 cm, 5 cm x 3 cm, 7 cm x 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. There are exactly 15 small cubes with no paint on any face.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:
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Answer: (A) 1 onlyExplanation
Unpainted cubes lie strictly inside: 5 x 3 x 1 = 15, so statement 1 is true.
Blue and green faces meet along 4 edges of length 3 cm.
Each such edge has 3 cubes, of which the 2 end cubes also touch a red face, leaving 1 cube per edge, i.e. 4 cubes.
Statement 2 says 6, which is false.
Answer: option (a).Source: UPSC CSAT 2023 question; answer key checked by independent computation
- #4UPSC CSAT 2022There are eight equidistant points on a circle. How many right-angled triangles can be drawn using these points as vertices and taking the diameter as one side of the triangle?
- A24
- B16
- C12
- D8
Answer: (A) 24Explanation
Eight equally spaced points give 4 diameters.
Any third point on the circle makes a right angle opposite the diameter (angle in a semicircle). 6 points remain for each diameter.
4 x 6 = 24 triangles.
Answer: option (a).Source: UPSC CSAT 2022 question; answer key checked by independent computation
- #5UPSC CSAT 2022Consider the following statements in respect of a rectangular sheet of length 20 cm and breadth 8 cm:1. It is possible to cut the sheet exactly into 4 square sheets.2. It is possible to cut the sheet into 10 triangular sheets of equal area.Which of the above statements is/are correct?
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Answer: (C) Both 1 and 2Explanation
Statement 1: cut two 8 x 8 squares (16 cm of length), leaving a 4 x 8 strip, which splits into two 4 x 4 squares. That is exactly 4 squares.
Statement 2: cut the sheet into five 4 x 8 rectangles and cut each along a diagonal: 10 triangles of equal area (16 sq cm each).
Both are possible.
Answer: option (c).Source: UPSC CSAT 2022 question; answer key checked by independent computation
- #6UPSC CSAT 2021A bank employee drives 10 km towards South from her house and turns to her left and drives another 20 km. She again turns left and drives 40 km, then she turns to her right and drives for another 5 km. She again turns to her right and drives another 30 km to reach her bank where she works. What is the shortest distance between her bank and her house?
- A20 km
- B25 km
- C30 km
- D35 km
Answer: (B) 25 kmExplanation
Take house as origin: South 10 gives (0,-10). Left turn faces East: 20 km gives (20,-10).
Left again faces North: 40 km gives (20,30). Right turn faces East: 5 km gives (25,30). Right again faces South: 30 km gives (25,0).
Distance from origin = 25 km.
Answer: option (b).Source: UPSC CSAT 2021 question; answer key checked by independent computation
- #7UPSC CSAT 2021There are three points P, Q and R on a straight line such that PQ:QR = 3:5. If n is the number of possible values of PQ:PR, then what is n equal to?
- A1
- B2
- C3
- D4
Answer: (B) 2Explanation
Let PQ = 3k and QR = 5k.
If R is on the far side of Q from P: PR = 8k, ratio 3:8.
If R is on the same side as P: PR = 2k, ratio 3:2.
So there are 2 possible values.
Answer: option (b).Source: UPSC CSAT 2021 question; answer key checked by independent computation
- #8UPSC CSAT 2021A cubical vessel of side 1 m is filled completely with water. How many millilitres of water is contained in it (neglect thickness of the vessel)?
- A1000
- B10000
- C100000
- D1000000
Answer: (D) 1000000Explanation
Volume = 1 m^3 = 100 x 100 x 100 cm^3 = 10^6 cm^3.
Since 1 cm^3 = 1 ml, the vessel holds 1,000,000 ml.
Answer: option (d).Source: UPSC CSAT 2021 question; answer key checked by independent computation
- #9UPSC CSAT 2020Let x, y be the volumes; m, n be the masses of two metallic cubes P and Q respectively. Each side of Q is two times that of P and mass of Q is two times that of P. Let u = m/x and v = n/y. Which one of the following is correct?
- Au = 4v
- Bu = 2v
- Cv = u
- Dv = 4u
Answer: (A) u = 4vExplanation
If P has side a, then x = a^3 and y = 8a^3. Also n = 2m.
v = 2m/(8a^3) = (1/4)(m/a^3) = u/4.
So u = 4v.
Answer: option (a).Source: UPSC CSAT 2020 question; answer key checked by independent computation
- #10UPSC CSAT 2020Consider the following statements:1. The minimum number of points of intersection of a square and a circle is 2.2. The maximum number of points of intersection of a square and a circle is 8.Which of the above statements is/are correct?
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Answer: (B) 2 onlyExplanation
A circle can lie wholly inside or wholly outside a square, giving 0 intersections, so Statement 1 is false.
Each side can cut the circle at most twice, so at most 4 x 2 = 8 points; this is achievable. Statement 2 is true.
Answer: option (b).Source: UPSC CSAT 2020 question; answer key checked by independent computation
- #11UPSC CSAT 2019A solid cube is painted yellow, blue and black such that opposite faces are of same colour. The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are Big. None of the faces of the bigger cubes is painted blue. How many cubes have only one face painted?
- A4
- B6
- C8
- D10
Answer: (C) 8Explanation
Volume check: 32 small + 4 big of double side gives a 4 x 4 x 4 cube.
Big cubes cannot touch the blue faces, so they form the middle layer; the two blue faces are top and bottom, each with 32/2 = 16 unit cubes.
On each blue face only the central 2 x 2 = 4 unit cubes have just one painted face; big cubes show two painted faces each.
Total = 4 + 4 = 8.
Answer: option (c).Source: UPSC CSAT 2019 question; answer key checked by independent computation
- #12UPSC CSAT 2019P, Q and R are three towns. The distance between P and Q is 60 km, whereas the distance between P and R is 80 km. Q is in the West of P and R is in the South of P. What is the distance between Q and R?
- A140 km
- B130 km
- C10km
- D100 km
Answer: (D) 100 kmExplanation
West and South are perpendicular, so PQR is a right triangle at P.
QR = sqrt(60^2 + 80^2) = sqrt(10000) = 100 km.
Answer: option (d).Source: UPSC CSAT 2019 question; answer key checked by independent computation
- #13UPSC CSAT 2018How many diagonals can be drawn by joining the vertices of an octagon?
- A20
- B24
- C28
- D64
Answer: (A) 20Explanation
Lines joining any two of 8 vertices = 8C2 = 28.
Subtract the 8 sides: 28 - 8 = 20 diagonals.
Answer: option (a).Source: UPSC CSAT 2018 question; answer key checked by independent computation
- #14UPSC CSAT 2018A 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?
- A12
- B8
- C6
- D4
Answer: (A) 12Explanation
Cubes with exactly two painted faces lie on the edges, excluding corners.
Each edge of a 3x3x3 cube has 3 - 2 = 1 such cube.
There are 12 edges, so 12 cubes.
Answer: option (a).Source: UPSC CSAT 2018 question; answer key checked by independent computation
- #15UPSC CSAT 2018There 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?
- A4
- B6
- C8
- D12
Answer: (C) 8Explanation
An equilateral triangle needs vertices 24/3 = 8 steps apart.
Each starting point gives one triangle, but each triangle is counted three times.
Number = 24 / 3 = 8.
Answer: option (c).Source: UPSC CSAT 2018 question; answer key checked by independent computation
- #16UPSC CSAT 2018Twelve 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?
- A5/7 sq cm
- B7/5 sq cm
- C1 sq cm
- D25/12 sq cm
Answer: (C) 1 sq cmExplanation
With side s, the rectangle is 4s long and 3s wide.
Its diagonal = sqrt(16s^2 + 9s^2) = 5s = 5, so s = 1 cm.
Area of each square = 1 sq cm.
Answer: option (c).Source: UPSC CSAT 2018 question; answer key checked by independent computation
- #17UPSC CSAT 2017The outer surface of a 4 cm x 4 cm x 4 cm cube is painted completely in red. It is sliced parallel to the faces to yield sixty four 1 cm x 1 cm x 1 cm small cubes. How many small cubes do not have painted faces?
- A8
- B16
- C24
- D36
Answer: (A) 8Explanation
Cubes with no painted face form the inner core of side 4 - 2 = 2.
Count = 2 x 2 x 2 = 8.
Answer: option (a).Source: UPSC CSAT 2017 question; answer key checked by independent computation
- #18UPSC CSAT 2017Two walls and a ceiling of a room meet at right angles at a point P. A fly is in the air 1 m from one wall, 8 m from the other wall and 9 m from the point P. How many meters is the fly from the ceiling?
- A4
- B6
- C12
- D15
Answer: (A) 4Explanation
The fly's distances from the three mutually perpendicular surfaces are 1, 8 and x.
Distance from P: 1^2 + 8^2 + x^2 = 9^2, so x^2 = 16.
x = 4 m.
Answer: option (a).Source: UPSC CSAT 2017 question; answer key checked by independent computation
- #19UPSC CSAT 2016A piece of tin is in the form of a rectangle having length 12 cm and width 8 cm. This is used to construct a closed cube. The side of the cube is:
- A2 cm
- B3 cm
- C4 cm
- D6 cm
Answer: (C) 4 cmExplanation
Surface area of the closed cube = area of the tin = 12 x 8 = 96 cm^2.
6 x side^2 = 96, so side^2 = 16.
Side = 4 cm.
Answer: option (c).Source: UPSC CSAT 2016 question; answer key checked by independent computation
- #20UPSC CSAT 2016A person climbs a hill in a straight path from point 'O' on the ground in the direction of north-east and reaches a point 'A' after travelling a distance of 5 km. Then, from the point 'A' he moves to point 'B' in the direction of north-west. Let the distance AB be 12 km. Now, how far is the person away from the starting point 'O'?
- A7 km
- B13 km
- C17 km
- D11 km
Answer: (B) 13 kmExplanation
North-east and north-west are at right angles, so triangle OAB is right-angled at A.
OB = sqrt(5^2 + 12^2) = sqrt(169) = 13 km.
Answer: option (b).Source: UPSC CSAT 2016 question; answer key checked by independent computation
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