CAT Questions | CAT Geometry Questions

CAT Quantitative Aptitude | CAT Geometry Mensuration

Mensuration is a fairly intuitive topic in CAT Geometry. Other than knowing basic formulae, structural and spatial understanding of the shapes helps one nail Mensuration questions in CAT exam. Mensuration comprises of 2D and 3D shapes including (but not limited to) Spheres, Cubes, Cuboids, Cylinders, Cones and more. You can expect at least one question from Mensuration in CAT exam. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

  1. CAT Geometry: Mensuration - Data sufficiency

    A right circular cone has height H and radius R. A small cone is cut off at the top by a plane parallel to the base. At what height above the base the section has been made?
    Statement (I): H = 20 cm
    Statement (II): Volume of small cone: volume of large cone : 1:15

    1. If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone.
    2. If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together.
    3. If the question can be answered with either statement alone.
    4. If the question cannot be answered with the information provided.
    Choice B
    Can be answered only if both statements are used together.

  2. CAT Geometry: Mensuration - Sphere

    A sphere of radius r is cut by a plane at a distance of h from its center, thereby breaking this sphere into two different pieces. The cumulative surface area of these two pieces is 25% more than that of the sphere. Find h.

    1. r/√2
    2. r/√3
    3. r/√5
    4. r/√6
    Choice A
    r/√2

  3. CAT Geometry: Mensuration - Chords

    Two mutually perpendicular chords AB and CD meet at a point P inside the circle such that AP = 6 cms, PB = 4 units and DP = 3 units. What is the area of the circle?

    1. 125π/4 sq cms
    2. 100π/7 sq cms
    3. 125π/8 sq cms
    4. 52π/3 sq cms
    Choice A
    125π/4 sq cms

  4. CAT Geometry: Mensuration - Cylinders

    Cylindrical cans of cricket balls are to be packed in a box. Each can has a radius of 7 cm and height of 30 cm. Dimension of the box is l = 76 cm, b = 46 cm, h = 45 cm. What is the maximum number of cans that can fit in the box?

    1. 15
    2. 17
    3. 22
    4. 21
    Choice D
    21

  5. CAT Geometry: Mensuration - Triangles

    PQRS is a square of sides 2 cm & ST = 2 cm. Also, PT=RT. What is the area of ∆PST?
    Geometry - Mensuration: CAT Geometry Triangles Area

    1. 2 cm2
    2. √3 cm2
    3. √2 cm2
    4. 1/√2 cm2
    Choice C
    √2 cm2

  6. CAT Geometry: Mensuration - Length of string

    A string is wound around two circular disk as shown. If the radius of the two disk are 40 cm and 30 cm respectively. What is the total length of the string?
    Geometry - Mensuration: CAT Length of string

    1. 70 cm
    2. 70 + 165*π cm
    3. 70 + 120π cm
    4. 70 + 165*π/2 cm
    Choice D
    70 + 165*π/2 cm

  7. CAT Geometry: Mensuration - Cuboid

    Geometry - Mensuration: CAT Cuboid
    Figure above shows a box which has to be completely wrapped with paper. However, a single Sheet of paper need to be used without any tearing. The dimension of the required paper could be

    1. 17 cm by 4 cm
    2. 12 cm by 6 cm
    3. 15 cm by 4 cm
    4. 13 cm by 4 cm
    Choice B
    12 cm by 6 cm

  8. CAT Geometry: Mensuration - Right Circular Cone

    An inverted right circular cone has a radius of 9 cm. This cone is partly filled with oil which is dipping from a hole in the tip at a rate of 1cm3/hour. Currently the level of oil 3 cm from top and surface area is 36π cm2. How long will it take the cone to be completely empty?

    1. 72π hours
    2. 1 hour
    3. 3 hours
    4. 36π hours
    Choice A
    72π hours

  9. CAT Geometry: Mensuration - Square and Triangle

    A square PQRS has an equilateral triangle PTO inscribed as shown:
    Geometry - Mensuration: CAT Square and triangle
    What is the ratio of A∆PQT to A∆TRU?

    1. 1 : 3
    2. 1 : √3
    3. 1 : √2
    4. 1 : 2
    Choice D
    1 : 2

  10. CAT Geometry: Mensuration - Spheres and Cubes

    A spherical shaped sweet is placed inside a cube of side 5 cm such that the sweet just fits the cube. A fly is sitting on one of the vertices of the cube. What is the shortest distance the fly must travel to reach the sweet?

    1. 2.5 cm
    2. 5(√3 – 1) cm
    3. 5(√2 – 1) cm
    4. 2.5(√3 – 1) cm
    Choice D
    2.5(√3 – 1) cm

  11. CAT Geometry: Mensuration - Square shape backyard

    Anil grows tomatoes in his backyard which is in the shape of a square. Each tomato takes 1 cm2 in his backyard. This year, he has been able to grow 131 more tomatoes than last year. The shape of the backyard remained a square. How many tomatoes did Anil produce this year?

    1. 4225
    2. 4096
    3. 4356
    4. Insufficient Data
    Choice C
    4356

  12. CAT Geometry: Mensuration - Circles

    PQRS is a circle and circles are drawn with PO, QO, RO and SO as diameters areas A and B are shaded A/B is equal to
    Geometry - Mensuration: CAT Circles

    1. π
    2. 1
    3. π/4
    4. 2
    Choice B
    1

  13. CAT Geometry: Mensuration - Square

    ABCD is a square drawn inside a square PQRS of sides 4 cm by joining midpoints of the sides PQ, QR, RS, SP. Another square is drawn inside ABCD similarly. This process is repeated infinite number of times. Find the sum of all the squares.

    1. 16 cm2
    2. 28 cm2
    3. 32 cm2
    4. Infinite
    Choice C
    32 cm2

  14. CAT Geometry: Mensuration - Pentagon and Circles

    PQRST is a pentagon in which all the interior angles are unequal. A circle of radius ‘r’ is inscribed in each of the vertices. Find the area of portion of circles falling inside the pentagon.

    1. πr2
    2. 1.5πr2
    3. 2πr2
    4. 1.25πr2
    Choice B
    1.5πr2

  15. CAT Geometry: Mensuration - Circles

    Three circles with radius 2 cm touch each other as shown :-
    Geometry - Mensuration: CAT Circles
    Find the area of the circle, circumscribing the above figure.

    1. 3π(4+√3)2
    2. π/2(4+2√3)2
    3. π/4(4+2√3)22
    4. 12π−π/2(4+2√3)2
    Choice C
    π/4(4+2√3)22

  16. CAT Geometry: Mensuration - Concentric Circles

    There are 5 concentric circles that are spaced equally from each other by 1.25 cms. The innermost circle has a square of side √(32) cm inscribed in it. If a square needs to be inscribed in the outermost circle, what will be its area?

    1. 324 sq. cm.
    2. (66 + 40√2) sq. cm
    3. 210.125 sq. cm.
    4. 162 sq.cm.
    Choice D
    162 sq.cm.

  17. CAT Geometry: Mensuration - Cross section of sphere

    A spherical rubber ball of radius 14 cm is cut by a knife at a distance of “x” cm from its centre, into 2 different pieces. What should be the value of “x” such that the cumulative surface area of the newly formed pieces is 3/28 more than the rubber ball’s original surface area?

    1. 11.4 cm
    2. 3 cm
    3. 12.4 cm
    4. 7.5 cm
    Choice C
    12.4 cm

  18. The following questions are from IPMAT Rohtak and Indore sample papers. If you want to take these questions as a mock please click below.

    IPMAT Rohtak Sample Paper Mock
    IPMAT Indore Sample Paper Mock

    Please note that the explanation button will take you to the IPMAT solution page.


    Use the figure below to answer questions 18 through 19.

    IPM Rohtak Quants : Circles
  19. IPMAT 2020 Sample Paper - IPM Rohtak Quants,Mensuration

    What is the area of the shaded figure?

    1. 56.25π square feet
    2. 112.5π square feet
    3. 225π square feet
    4. 337.4π square feet
    Choice B
    112.5 \\pi\\)

  20. IPMAT 2020 Sample Paper - IPM Rohtak Quants,Mensuration

    What is the ratio of the area of Circle M and the area of Circle K?

    1. 1 : 8
    2. 1 : 4
    3. 1 : 2
    4. 1 : 1
    Choice B
    1:4

  21. IPMAT 2020 Sample Paper - IPM Rohtak Quants,Mensuration

    A room has floor size of 15*6sq.cm. What is the height of the room , if the sum of the areas of the base and roof is equal to the sum of the areas of the four walls ?

    1. 1.12cm
    2. 3.24cm
    3. 4.29cm
    4. 2.5cm
    Choice A
    1.12cm

  22. IPMAT 2020 Sample Paper - IPM Rohtak Quants,Mensuration

    A circle is inscribed in an equilateral triangle of side 24 cm, touching its sides. What is the area of
    the remaining portion of the triangle?

    1. 144√3 - 48π cm2
    2. 121√3 - 36π cm2
    3. 144√3 - 36π cm2
    4. 121√3 - 48π cm2
    Choice A
    144√3 - 48π cm2

  23. IPMAT 2019 Question Paper - IPM Indore Quants,Mensuration

    The number of whole metallic tiles that can be produced by melting and recasting a circular metallic plate, if each of the tiles has a shape of a right-angled isosceles triangle and the circular plate has a radius equal in length to the longest side of the tile (Assume that the tiles and plate are of uniform thickness, and there is no loss of material in the melting and recasting process) is

    12

  24. IPMAT 2019 Question Paper - IPM Indore Quants,Mensuration

    Points P, Q, R and S are taken on sides AB, BC, CD and DA of square ABCD respectively, so that AP : PB = BQ : QC = CR : RD = DS : SA = 1 : n . Then the ratio of the area of PQRS to the area of ABCD is

    1. 1 : (1 + n)
    2. 1 : n
    3. 1 + n2 : (1 + n)2
    4. (1 + n ) : (1 + n2)
    Choice C
    1 + n2 : (1 + n)2

  25. IPMAT 2019 Question Paper - IPM Indore Quants,Mensuration

    Three cubes with integer edge lengths are given. It is known that the sum of their surface areas is 564 cm2 Then the possible values of the sum of their volumes are

    1. 764 cm3 and 586 cm3
    2. 586 cm3and 564 cm3
    3. 764 cm3 and 564 cm3
    4. 586 cm3 and 786 cm3
    Choice A
    764 cm3 and 586 cm3

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.


  1. CAT 2017 Question Paper Slot 1 - Mensuration & Geometry

    A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to:

    1. 10
    2. 50
    3. 60
    4. 20
    Choice B
    50

  2. CAT 2017 Question Paper Slot 1 - Mensuration & Geometry

    A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is 3 cm, while its volume is 9 π cm3. Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is: (TITA)

    6

  3. CAT 2017 Question Paper Slot 2 - Geometry

    Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

    1. 3√2
    2. 3
    3. 4
    4. √3
    Choice B
    3

  4. CAT 2017 Question Paper Slot 2 - Geometry

    The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length. The perpendicular distance between the parallel sides of the trapezium is 12 cm. If the height of the pillar is 20 cm, then the total area, in sq cm, of all six surfaces of the pillar is

    1. 1300
    2. 1340
    3. 1480
    4. 1520
    Choice C
    1480

  5. CAT 2017 Question Paper Slot 2 - Geometry

    If three sides of a rectangular park have a total length 400 ft., then the area of the park is maximum when the length (in ft.) of its longer side is [TITA]

    200

  6. CAT 2018 Question Paper Slot 1 - Mensuration

    A right circular cone, of height 12 ft, stands on its base which has diameter 8 ft. The tip of the cone is cut off with a plane which is parallel to the base and 9 ft from the base. With π = 22/7, the volume, in cubic ft, of the remaining part of the cone is:[TITA]

    198

  7. CAT 2018 Question Paper Slot 1 - Mensuration

    In a parallelogram ABCD of area 72 sq cm, the sides CD and AD have lengths 9 cm and 16 cm, respectively. Let P be a point on CD such that AP is perpendicular to CD. Then the area, in sq cm, of triangle APD is

    1. 18√3
    2. 24√3
    3. 32√3
    4. 12√3
    Choice C
    32√3

  8. CAT 2018 Question Paper Slot 2 - Mensuration

    From a rectangle ABCD of area 768 sq cm, a semicircular part with diameter AB and area 72π sq cm is removed. The perimeter of the leftover portion, in cm, is

    1. 88 + 12π
    2. 80 + 16π
    3. 86 + 8π
    4. 82 + 24π
    Choice A
    88 + 12π

  9. CAT 2018 Question Paper Slot 2 - Mensuration

    The area of a rectangle and the square of its perimeter are in the ratio 1 : 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio

    1. 3 : 8
    2. 2 : 9
    3. 1 : 4
    4. 1 : 3
    Choice C
    1 : 4

  10. CAT 2018 Question Paper Slot 2 - Mensuration

    A parallelogram ABCD has area 48 sqcm. If the length of CD is 8 cm and that of AD is s cm, then which one of the following is necessarily true?

    1. s ≥ 6
    2. s ≠ 6
    3. 5 ≤ s ≤ 7
    4. s ≤ 6
    Choice A
    s ≥ 6

  11. CAT 2019 Question Paper Slot 1 - Mensuration

    Corners are cut off from an equilateral triangle T to produce a regular hexagon H. Then, the ratio of the area of H to the area of T is 

    1. 5 : 6
    2. 3 : 4
    3. 2 : 3
    4. 4 : 5
    Choice C
    2 : 3

  12. CAT 2019 Question Paper Slot 1 - Mensuration

    If the rectangular faces of a brick have their diagonals in the ratio 3 : 2 \\sqrt{3}) : \\sqrt{15}), then the ratio of the length of the shortest edge of the brick to that of its longest edge is

    1. 1 : \\sqrt{3})
    2. 2 : \\sqrt{5})
    3. \\sqrt{2}) :\\sqrt{3})
    4. \\sqrt{3}) : 2
    Choice A
    1 : \\sqrt{3})

  13. CAT 2019 Question Paper Slot 2 - Mensuration

    The base of a regular pyramid is a square and each of the other four sides is an equilateral triangle, length of each side being 20 cm. The vertical height of the pyramid, in cm, is

    1. 10\\sqrt{2})
    2. 8\\sqrt{3})
    3. 12
    4. 5\\sqrt{5})
    Choice A
    10\\sqrt{2})

  14. CAT 2019 Question Paper Slot 2 - Mensuration

    A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these has radius 3 cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

    1. 1044(4 + π)
    2. 8464π
    3. 928π
    4. 1026(1 + π)
    Choice D
    1026(1 + π)


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