Quantitative Aptitude

Mensuration

2,507 Questions
All Direct Formula Application 1067 Shape Transformations 343 Mensuration Ratio Equations 277 Combined & Hollow Solids 247 Successive Percentage Changes 178 Inscribed & Circumscribed Figures 165 Cost Estimation Problems 100 Liquid Pour & Submergence 67 Unit Conversions 35 Path & Border Calculations 17 Time & Work Integration 7 Scale & Model Proportions 2

Mensuration Questions

Multiple choice

A solid cylinder of base radius 12 cm and height 15 cm is melted and recasted into n toys, each in the shape of a right circular cone of height 9 cm mounted on a hemisphere of radius 3 cm. The value of n is

  1. 27

  2. 64

  3. 48

  4. 54
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Volume of cylinder = pi * 12^2 * 15 = 2160 * pi. Volume of one toy = volume of cone + volume of hemisphere = (1/3 * pi * 3^2 * 9) + (2/3 * pi * 3^3) = 27 * pi + 18 * pi = 45 * pi. n = 2160 * pi / 45 * pi = 48.

Multiple choice

If a square with each side 'a' is joined from opposite sides to form a cylinder, then area of each circular end (ignoring units) is

  1. a2

  2. a2/4π

  3. a2/π

  4. a2/4
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

When a square of side 'a' is joined at opposite sides to form a cylinder, the height of the cylinder is 'a' and the circumference of the circular base is 'a'. Circumference = 2*pi*r = a, so r = a / (2*pi). Area of circular end = pi*r^2 = pi * (a / (2*pi))^2 = pi * (a^2 / (4*pi^2)) = a^2 / (4*pi).

Multiple choice

A conical vessel of radius 12 cm and height 16 cm is filled with water. A sphere is lowered into water and its size is such that it touches the sides of the vessel and it is just immersed. What is the radius of the sphere?

  1. 5 cm

  2. 6 cm

  3. 6.5 cm

  4. 7 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using similar triangles in the cross-section, let r be the sphere radius. The distance from the cone vertex to the sphere center is 16 - r. By similarity, r / (16 - r) = 12 / sqrt(12^2 + 16^2) = 12 / 20 = 3/5. Solving 5r = 48 - 3r gives 8r = 48, so r = 6 cm.

Multiple choice

A closed box is built of wood of uniform thickness. Its external dimensions are 12 cm, 10 cm and 8 cm. If the inner surface area is 376 square cm, then what is the thickness of the wood?

  1. 0.5 cm

  2. 1 cm

  3. 1.5 cm

  4. 2 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let thickness be x. Inner dimensions are (12-2x), (10-2x), (8-2x). Inner surface area = 2 * [(12-2x)(10-2x) + (10-2x)(8-2x) + (12-2x)(8-2x)] = 376. Dividing by 2: (120 - 44x + 4x^2) + (80 - 36x + 4x^2) + (96 - 40x + 4x^2) = 188. 12x^2 - 120x + 296 = 188. 12x^2 - 120x + 108 = 0. x^2 - 10x + 9 = 0. (x-1)(x-9) = 0. x=1.

Multiple choice

The largest sphere is curved out of a cube of side 7 cm. Find the volume of the sphere (Use π=22/7)

  1. 179.65 cm3

  2. 179.64 cm3

  3. 179.66 cm3

  4. 179.68 cm3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The largest sphere that can be carved out of a cube has a diameter equal to the side of the cube. Here, the diameter is 7 cm, so the radius is 3.5 cm. The volume is (4/3) * pi * r^3 = (4/3) * (22/7) * (3.5)^3 = (4/3) * (22/7) * (7/2)^3 = (4/3) * (22/7) * (343/8) = 179.666... cm^3.

Multiple choice

The surface area of a cube of length x is equal to the surface area of a sphere of radius y. Consider the following statements: 1. 2x > 3y 2. The volume of the cube is greater than the volume of the sphere. Which of the above statements is/are correct?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Surface area of cube = 6x^2. Surface area of sphere = 4 * pi * y^2. 6x^2 = 4 * pi * y^2 implies x^2 = (2/3) * pi * y^2, so x = y * sqrt(2 * pi / 3) approx 1.447y. Thus 2x approx 2.89y, so 2x > 3y is false. Volume of cube = x^3 approx 3.03y^3. Volume of sphere = (4/3) * pi * y^3 approx 4.18y^3. So volume of cube is not greater than volume of sphere. Both statements are false.

Multiple choice

A solid metallic cuboid of dimensions 18 cm × 36 cm × 72 cm is melted and recast into 8 cubes of the same volume. What is the ratio of the total surface area of the cuboid to the sum of the lateral surface areas of all 8 cubes?

  1. 4 : 7

  2. 7 : 8

  3. 7 : 12

  4. 2 : 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Volume of cuboid = 18 * 36 * 72 = 46656. Volume of each cube = 46656 / 8 = 5832. Side of cube = cube root of 5832 = 18. Surface area of cuboid = 2(18*36 + 36*72 + 72*18) = 2(648 + 2592 + 1296) = 2(4536) = 9072. Lateral surface area of one cube = 4 * side^2 = 4 * 18^2 = 4 * 324 = 1296. Total lateral area of 8 cubes = 8 * 1296 = 10368. Ratio = 9072 / 10368 = 7/8.

Multiple choice

A candle in the shape of cylinder is manufactured in such a way that its height is 3 inches more than its radius. Which of the following expresses the volume, V, in cubic inches, of the candle, where r is the radius, in inches?

  1. 3πr2 + πr

  2. 3πr + πr3

  3. 3πr2 + πr3

  4. 3πr3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The volume of a cylinder is V = pi * r^2 * h. Given h = r + 3, substitute this into the formula: V = pi * r^2 * (r + 3) = pi * r^3 + 3 * pi * r^2.

Multiple choice

A sample of pine has a density of 620 kilograms per cubic meter. The sample is in the shape of a cylinder, where the base has a radius of 0.75 meters and a height of 1.5 meters. To the nearest whole number, what is the mass, in kilograms, of this sample?

  1. 620 kg

  2. 1,240 kg

  3. 1,643 kg

  4. 1,781 kg

  5. a
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Volume of cylinder = pi * r^2 * h = 3.14159 * (0.75)^2 * 1.5 = 3.14159 * 0.5625 * 1.5 = 2.6507 cubic meters. Mass = density * volume = 620 * 2.6507 = 1643.43 kg. Rounded to the nearest whole number, it is 1643 kg.