CAT Questions | CAT Geometry Questions

CAT Quantitative Aptitude | CAT Geometry Triangles , Circles

A CAT Geometry question that appears in the Quantitative Aptitude section of the CAT Exam broadly tests an aspirant on the concepts - Triangles, Circles, Quadrliaterals, Polygons & mixture of the above mentioned concepts. In CAT Exam, one can generally expect to get 4~6 questions from CAT Geometry. CAT Geometry is an important topic with lots of weightage in the 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: Obtuse Triangles - Integers

    x, y, z are integer that are side of an obtuse-angled triangle. If xy = 4, find z.

    1. 2
    2. 3
    3. 1
    4. More than one possible value of z exists
    Choice B
    3

  2. CAT Geometry: Isosceles Triangles - Integers

    How many isosceles triangles with integer sides are possible such that sum of two of the side is 12?

    1. 11
    2. 6
    3. 17
    4. 23
    Choice C
    17

  3. CAT Geometry: Triangles - Area

    Sides of a triangle are 6, 10 and x for what value of x is the area of the △ the maximum?

    1. 8 cms
    2. 9 cms
    3. 12 cms
    4. None of these
    Choice D
    None of these

  4. CAT Geometry: Equilateral triangle and circle

    Two circles are placed in an equilateral triangle as shown in the figure. What is the ratio of the area of the smaller circle to that of the equilateral triangle
    Geometry - triangles: CAT Geometry Triangles Area

    1. π:36√3
    2. π:18√3
    3. π:27√3
    4. π:42√3
    Choice C
    π:27√3

  5. CAT Geometry: Triangles - Perimeter

    Perimeter of a △ with integer sides is equal to 15. How many such triangles are possible?

    1. 7
    2. 6
    3. 8
    4. 5
    Choice A
    7

  6. CAT Geometry: Equilateral Triangle and Square

    There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral △. What is the ratio of the area of the square to that of the equilateral triangle?

    1. 12 : 12 + 7√3
    2. 24 : 24 + 7√3
    3. 18 : 12 + 15√3
    4. 6 : 6 + 5√3
    Choice A
    12 : 12 + 7√3

  7. CAT Geometry: Triangles - Integers

    △ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible?

    1. 8
    2. 6
    3. 9
    4. 12
    Choice C
    9

  8. CAT Geometry: Regular Polygon

    ABCDE is a regular pentagon. O is a point inside the pentagon such that AOB is an equilateral triangle. What is ∠OEA?
    CAT Question - Geometry - Regular Polygon

    1. 66°
    2. 48°
    3. 54°
    4. 72°
    Choice A
    66°

  9. CAT Geometry: Triangles Properties

    △ has sides a2, b2 and c2. Then the triangle with sides a, b, c has to be:

    1. Right-angled
    2. Acute-angled
    3. Obtuse-angled
    4. Can be any of these three
    Choice B
    Acute-angled

  10. CAT Geometry: Right Triangle Properties

    Consider a right–angled triangle with inradius 2 cm and circumradius of 7 cm. What is the area of the triangle?

    1. 32 sq cms
    2. 31.5 sq cms
    3. 32.5 sq cms
    4. 33 sq cms
    Choice A
    32 sq cms

  11. CAT Geometry: Regular Polygon

    What is the ratio of longest diagonal to the shortest diagonal in a regular octagon?

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

  12. CAT Geometry: Altitude of a Triangle

    Find the altitude to side AC of triangle with side AB = 20 cm, AC = 20 cm, BC = 30 cm.

    1. 10√7
    2. 8√7
    3. 7.5√7
    4. 15√7
    Choice C
    7.5√7

  13. CAT Geometry: Regular Polygon

    ABCDEF is a regular hexagon inscribed inside a circle. If the shortest diagonal of the hexagon is of length 3 units, what is the area of the shaded region.
    CAT Question - Geometry - Regular Octagon

    1. 1/6(3π − (9√3)/2)
    2. 1/6(2π − (6√3)/2)
    3. 1/6(3π − (8√3)/2)
    4. 1/6(6π − (15√3)/2)
    Choice A
    1/6(3π − (9√3)/2)

  14. CAT Geometry: Circles

    A circle of radius 5 cm has chord RS at a distance of 3 units from it. Chord PQ intersects with chord RS at T such that TS = 1/3 of RT. Find minimum value of PQ.

    1. 6√3
    2. 4√3
    3. 8√3
    4. 2√3
    Choice B
    4√3

  15. CAT Geometry: Triangles - Area

    Triangle has perimeter of 6 + 2√3 . One of the angles in the triangle is equal to the exterior angle of a regular hexagon another angle is equal to the exterior angle of a regular 12-sided polygon. Find area of the triangle.

    1. 2√3
    2. √3
    3. √3/2
    4. 3
    Choice A
    2√3

  16. CAT Geometry: Quadrilateral

    Area of a Rhombus of perimeter 56 cms is 100 sq cms. Find the sum of the lengths of its diagonals

    1. 33.40
    2. 34.40
    3. 31.20
    4. 32.30
    Choice B
    34.40

  17. CAT Geometry: Quadrilateral

    Rhombus has a perimeter of 12 and one angle = 120°. Find its area.

    1. 9 * (√3)/2
    2. 3 * (√3)/2
    3. 9 * √3
    4. 18 * √3
    Choice A
    9 * (√3)/2

  18. CAT Geometry: Triangles - Area

    Circle with center O and radius 25 cms has a chord AB of length of 14 cms in it. Find the area of triangle AOB

    1. 144 cm2
    2. 121 cm2
    3. 156 cm2
    4. 168 cm2
    Choice D
    168 cm2

  19. CAT Geometry: Circles

    Two mutually perpendicular chords AB and CD intersect at P. AP = 4, PB = 6, CP = 3. Find radius of the circle.

    1. 31.25(1/2)
    2. 37.5(1/2)
    3. 26(1/2)
    4. 52(1/2)
    Choice A
    31.25(1/2)

  20. CAT Geometry: Triangles

    Triangle ABC has angles A = 60° and B = 70°. The incenter of this triangle is at I. Find angle BIC.

    1. 90°
    2. 130°
    3. 80°
    4. 120°
    Choice D
    120°

  21. CAT Geometry: Quadrilateral

    Rhombus of side 6 cm has an angle equal to the external angle of a regular octagon. Find the area of the rhombus.

    1. 18√2 cm2
    2. 9√2 cm2
    3. 15√2 cm2
    4. 12√2 cm2
    Choice A
    18√2 cm2

  22. CAT Geometry: Circle, Square and Triangle

    A circle inscribed in a square of side 2 has an equilateral triangle inscribed inside it. What is the ratio of areas of the equilateral triangle to that of the square?

    1. 9√3 : 16
    2. 3√3 : 4
    3. 9√3 : 4
    4. 3√3 : 16
    Choice D
    3√3 : 16

  23. CAT Geometry: Isosceles Triangles

    An acute-angled isosceles triangle has two of its sides equal to 10 and 16. Find the area of this triangle.

    1. √231 units
    2. 12√66 units
    3. 24 units
    4. 5√231 units
    Choice D
    5√231 units

  24. CAT Geometry: Triangles - Area

    Three equal circles are placed inside an equilateral triangle such that any circle is tangential to two sides of the equilateral triangle and to two other circles. What is the ratio of the areas of one circle to that of the triangle?

    1. π : (6+4√3)
    2. 3π : (6+4√3)
    3. 2π : (6+4√3)
    4. π : (6+2√3)
    Choice A
    π : (6+4√3)

  25. CAT Geometry: Triangle and Square

    There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral △. What is the ratio of the area of the square to that of the equilateral triangle?

    1. √3 : (5 + 4√3)
    2. 2√3 : (7 + 4√3)
    3. 4√3 : (7 + 4√3)
    4. 4√3 : (5 + 2√3)
    Choice C
    4√3 : (7 + 4√3)

  26. CAT Geometry: Squares and Circles

    Consider Square S inscribed in circle C, what is the ratio of the areas of S and C? And, Consider Circle Q inscribed in Square S, what is the ratio of the areas of S and Q?

    1. 2:π, 4:π
    2. 4:π, 2:π
    3. 1:π, 4:π
    4. 2:π, 1:π
    Choice A
    2:π, 4:π

  27. CAT Geometry: Triangles - Area

    Consider equilateral triangle T inscribed in circle C, what is ratio of the areas of T and C? Consider Circle C inscribed in equilateral triangle T, what is ratio of the areas of T and C?

    1. 3√3:π , 3√3:16π
    2. 3√3:4π , 3√3:π
    3. √3:π , 3√3:4π
    4. √3:π , √3:16π
    Choice B
    3√3:4π , 3√3:π

  28. CAT Geometry: Regular Octagon

    Consider Regular Hexagon H inscribed in circle C, what is ratio of the areas of H and C? Consider Circle C inscribed in Regular Hexagon H, what is ratio of the areas of H and C?

    1. 2√3 : 3π , 3√3 : 4π
    2. 3√3 : π , 3√3 : 4π
    3. 3√3 : 2π, 2√3 : π
    4. √3 : π , √3 : 4π
    Choice C
    3√3 : 2π, 2√3 : π

  29. CAT Geometry: Triangles

    What is the distance between the orthocentre and the circumcenter of a triangle who sides measure 24 cm, 26 cm and 10 cm?

    1. 13 cm
    2. 12 cm
    3. 7.5 cm
    4. √30 cm
    Choice A
    13 cm

  30. CAT Geometry: Circles and Triangles

    Two circles with centres O1 and O2 touch each other externally at a point R. AB is a tangent to both the circles passing through R. P’Q’ is another tangent to the circles touching them at P and Q respectively and also cutting AB at S. PQ measures 6 cm and the point S is at distance of 5 cms and 4 cms from the centres of the circles. What is the area of the triangle SO1O2?

    1. 9 cm2
    2. 3(4+√7)/2 cm2
    3. 27/2 cm2
    4. (3√41)/2 cm2
    Choice B
    3(4+√7)/2 cm2

  31. CAT Geometry: Circles

    What is the circumference of the below circle given that AB is the diameter and XY is perpendicular to AB?
    CAT Question - Geometry - circles

    1. 8π cm
    2. π√34 cm
    3. 34π/3 cm
    4. π√31/3 cm
    Choice C
    34π/3 cm

  32. CAT Geometry: Circles

    Find ∠PRB. Given
    I. ∠BPQ = 22 and O is the centre of the circle
    II. ∠RBP = 54 and chord PQ is parallel to AB
    CAT Question - Geometry - circles

    1. Either I or II individually is sufficient
    2. Both I and II together are required
    3. One of the statements alone is sufficient
    4. Need more data
    Choice B
    Both I and II together are required

  33. CAT Geometry: Triangles

    The two sides of a triangle are 8 cm and 9 cm and one angle is 60. Which of the following can be the length of its third side?
    I. √23 cm
    II. √73 cm
    III.(4.5 - √3.25) cm
    IV. (4 + √33) cm
    V. (9 + √13) cm

    1. Only II and IV
    2. Only I and III
    3. Only I, II and V
    4. Only II, III and IV
    Choice D
    Only II, III and IV

  34. CAT Geometry: Quadrilateral

    There is a set of parallel lines with x lines in it and another set of parallel lines with y lines in it. The lines intersect at 12 points. If x > y, find the maximum number of parallelograms that can be formed.

    1. 16
    2. 15
    3. 18
    4. 33
    Choice C
    18

  35. CAT Geometry: Polygon and Circles

    There are 2 concentric circles, one big and one small. A square ABCD is inscribed inside the big circle while the same square circumscribes the small circle. The square touches the small circle at points P, Q, R and S. Determine the ratio of circumference of big circle to the polygon PQRS.

    1. π : 2
    2. 2 : π
    3. 2 : √2
    4. π : √2
    Choice A
    π : 2

  36. CAT Geometry: Quadrilaterals

    If PQ || RS, find the value of x
    CAT Question - Geometry - Triangles

    1. 7
    2. 3
    3. Both A & B
    4. None of these
    Choice C
    Both A & B

  37. CAT Geometry: Circles

    A circle is inscribed in a semi-circle as shown:-
    CAT Question - Geometry - circles
    The radius of the circle possibly is:-

    1. (√2 + 1)/2
    2. √2 − 1/2
    3. 1 − √2
    4. √2 − 1
    Choice D
    √2 − 1

  38. CAT Geometry: Circles

    Two circles of radius 5 cm have a direct tangent PQ and an indirect tangent RS. Find the length of PQ if RS = 24 cm.

    1. 29 cm
    2. 13 cm
    3. 26 cm
    4. Cannot be determined due to lack of information
    Choice C
    26 cm

  39. CAT Geometry: Circles and Triangles

    In the below figure which of the following holds good?
    CAT Question - Geometry - circles

    1. ∠SQO = ∠ROP
    2. 2 ∠ROP = ∠SOR
    3. ∠POR = ∠ASO
    4. ∠QOX’ = ∠SOR + ∠ROP
    Choice A
    ∠SQO = ∠ROP

  40. CAT Geometry: Circles and Triangles

    In the below figure, If \\frac{PR}{PQ}\\) = \\frac{PQ}{RQ}\\), then

    CAT Question - Geometry - Lines
    1. \\frac{PQ}{QR}\\) > 2
    2. \\frac{PQ}{QR}\\) = 2
    3. \\frac{PQ}{QR}\\) < 2
    4. can’t be determined
    Choice C
    \\frac{PQ}{QR}\\) = 2

  41. CAT Geometry: Triangles

    A right angled triangle PQR is such that ∠PRQ = 90° and QR = 4 cm T is a point on QR such that PT = 3 cm, and perimeter of triangle PQT = Perimeter of triangle PTR Then, QT/TR takes the value:

    1. QT/TR < 1/3
    2. 1/3 < QT/TR < 1
    3. QT/TR > 1
    4. can’t be determined
    Choice B
    1/3 < QT/TR < 1

  42. CAT Geometry: Triangles

    M and N are two points on the side PQ and PR of a triangle PQR respectively such that MNQR is a trapezium and MN:QR = 2:5. Find the ratio of the area of triangle PMN : Trapezium MNQR.

    1. 4:25
    2. 1:2
    3. 4:21
    4. 4:10
    Choice C
    4:21

  43. CAT Geometry: Triangles

    In the below figure, ΔABC is right angled and AC = 100 cm. Also, AD = DE = EF = FC. Find the value of: BD2 + BE2 + BF2 (in cm2)

    CAT Question - Geometry - Triangles
    1. 10,000 cm2
    2. 5,000 cm2
    3. 8,750 cm2
    4. 12,500 cm2
    Choice C
    8,750 cm2

  44. CAT Geometry: Triangles

    The Olympics committee came up with a new rule. The flag of the gold medal winning team would be hoisted to the right (AB) at 5m. The flag of silver medal winning team would be hoisted to the left (PQ) at a height of 3m. The flag (MN) of bronze medal winning team would be hoisted at the point of intersection of the line joining the top of each of AB and PQ to the foot of other, as shown in the figure below. A and P are 8m apart. In a wrestling event, India won the bronze medal. Find the height at which the Indian flag was hoisted.
    CAT Question - Geometry - Triangles

    1. 2 m
    2. \\frac{5}{2}\\) m
    3. \\frac{5}{8}\\) m
    4. \\frac{15}{8}\\) m
    Choice D
    \\frac{15}{8}\\) m

  45. CAT Geometry: Polygons

    The number of sides in a regular polygon is ‘T’ times the number of diagonals in it. What is the interior angle of this polygon in terms of T?

    1. 180 * \\frac{(T+2)}{(3T+2)}\\)
    2. 540 * \\frac{(T+2)}{(3T+2)}\\)
    3. 360 * \\frac{(T+2)}{(3T+2)}\\)
    4. 90 * \\frac{(T+2)}{(3T+2)}\\)
    Choice A
    180 * \\frac{(T+2)}{(3T+2)}\\)

  46. 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.


  47. IPMAT 2019 Question Paper - IPM Indore Quants,Geometry

    The sum of the interior angles of a convex n-sided polygon is less than \2019^{\circ}\\). The maximum possible value of n is

    13

  48. IPMAT 2019 Question Paper - IPM Indore Quants,Geometry

    A chord is drawn inside a circle, such that the length of the chord is equal to the radius of the circle. Now, two circles are drawn, one on each side of the chord, each touching the chord at its midpoint and the original circle. Let k be the ratio of the areas of the bigger inscribed circle and the smaller inscribed circle, then k equals

    1. (2 + √3)
    2. (1 + √2)
    3. (7 + 4√3)
    4. (97 + 56√3)
    Choice D
    (97 + 56√3)

  49. IPMAT 2019 Question Paper - IPM Indore Quants,Geometry

    On a circular path of radius 6 m a boy starts from a point A on the circumference and walks along a chord AB of length 3 m. He then walks along another chord BC of length 2 m to reach point C. The point B lies on the minor arc AC. The distance between point C from point A is

    1. \\frac{\sqrt{15} + \sqrt{35}}{2}\\) m
    2. 8 m
    3. √13 m
    4. 6 m
    Choice A
    \\frac{\sqrt{15} + \sqrt{35}}{2}\\) m

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 - Geometry

    From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC. The area, in sq ft, of the remaining portion of triangle ABC is:

    1. 225√3
    2. \\frac{500}{√3}\\)
    3. \\frac{275}{√3}\\)
    4. \\frac{250}{√3}\\)
    Choice B
    \\frac{500}{√3}\\)

  2. CAT 2017 Question Paper Slot 1 - Geometry

    Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC. Let BPC be an arc of a circle centered at A and lying between BC and BQC. If AB has length 6 cm then the area, in sq. cm, of the region enclosed by BPC and BQC is:

    1. 9π - 18
    2. 18
    3. 9
    Choice B
    18

  3. CAT 2017 Question Paper Slot 1 - Geometry

    Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively. The minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour is: (TITA)

    24

  4. CAT 2017 Question Paper Slot 2 - Geometry

    ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is [TITA]

    90

  5. CAT 2017 Question Paper Slot 2 - Geometry

    Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB. If the perpendicular distance of P from each of AB, BC, and CA is 4(√2 - 1) cm, then the area, in sq. cm, of the triangle ABC is [TITA]

    16

  6. CAT 2018 Question Paper Slot 1 - Geometry

    In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is

    1. \\frac{π}{4})\\frac{1}{2})
    2. \\frac{π}{6})\\frac{1}{2})
    3. \\frac{π}{4√3})\\frac{1}{2})
    4. \\frac{π}{3√3})\\frac{1}{2})
    Choice D
    \\frac{π}{3√3})\\frac{1}{2})

  7. CAT 2018 Question Paper Slot 1 - Geometry

    Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will be

    1. 192√3
    2. 164√3
    3. 248√3
    4. 188√3
    Choice A
    192√3

  8. CAT 2018 Question Paper Slot 1 - Geometry

    Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

    1. 25 , 10
    2. 24 , 12
    3. 25 , 9
    4. 24 , 10
    Choice D
    24 , 10

  9. CAT 2018 Question Paper Slot 1 - Geometry

    Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square whose area is 62.5% of that of ABCD and CG is longer than EB, then the ratio of length of EB to that of CG is:

    1. 1 : 3
    2. 4 : 9
    3. 2 : 5
    4. 3 : 8
    Choice A
    1 : 3

  10. CAT 2018 Question Paper Slot 1 - Geometry

    In a circle, two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm. If the distance between these chords is 1 cm, then the radius of the circle, in cm, is

    1. √13
    2. √14
    3. √11
    4. √12
    Choice A
    √13

  11. CAT 2018 Question Paper Slot 2 - Geometry

    On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, is (TITA)

    24 cm

  12. CAT 2018 Question Paper Slot 2 - Geometry

    A chord of length 5 cm subtends an angle of 60° at the centre of a circle. The length, in cm, of a chord that subtends an angle of 120° at the centre of the same circle is

    1. 5√3
    2. 6√2
    3. 8
    Choice B
    5√3

  13. CAT 2019 Question Paper Slot 1 - Circles

    In a circle of radius 11 cm, CD is a diameter and AB is a chord of length 20.5 cm. If AB and CD intersect at a point E inside the circle and CE has length 7 cm, then the difference of the lengths of BE and AE, in cm, is

    1. 1.5
    2. 3.5
    3. 0.5
    4. 2.5
    Choice C
    0.5

  14. CAT 2019 Question Paper Slot 1 - Geometry

    AB is a diameter of a circle of radius 5 cm. Let P and Q be two points on the circle so that the length of PB is 6 cm, and the length of AP is twice that of AQ. Then the length, in cm, of QB is nearest to

    1. 8.5
    2. 9.3
    3. 9.1
    4. 7.8
    Choice C
    9.1

  15. CAT 2019 Question Paper Slot 2 - Geometry

    In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq cm, is

    1. 80
    2. 68
    3. 72
    4. 78
    Choice C
    72

  16. CAT 2019 Question Paper Slot 2 - Geometry

    Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is 

    1. \\frac{π}{3})
    2. 1
    3. \\frac{1}{\sqrt{2}})
    4. \\sqrt{2})
    Choice B
    1

  17. CAT 2019 Question Paper Slot 2 - Geometry

    Let A and B be two regular polygons having a and b sides, respectively. If b = 2a and each interior angle of B is 3/2 times each interior angle of A, then each interior angle, in degrees, of a regular polygon with a + b sides is [TITA]

    150

  18. CAT 2019 Question Paper Slot 2 - Geometry

    Let ABC be a right-angled triangle with hypotenuse BC of length 20 cm. If AP is perpendicular on BC, then the maximum possible length of AP, in cm, is

    1. 10
    2. 8\\sqrt{2})
    3. 6\\sqrt{2})
    4. 5
    Choice A
    10


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