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 cylinder of radius $6\ cm$ and height $h$ cm is filled with ice cream. The ice cream is then distributed among $10$ children in identical cones having hemispherical top. The radius of the base of the cone is $3\ cm$ and its height is $12\ cm$. Then the height $h$ of the cylinder must be
  1. $\dfrac{100}{7}\ cm$
  2. $18\ cm$
  3. $15\ cm$
  4. $\dfrac{200}{11}\ cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Volume of cylinder = pi * 6^2 * h = 36 * pi * h. Volume of 10 cones = 10 * (1/3 * pi * 3^2 * 12 + 2/3 * pi * 3^3) = 10 * (36 * pi + 18 * pi) = 10 * 54 * pi = 540 * pi. 36 * pi * h = 540 * pi, so h = 540/36 = 15 cm.

Multiple choice

If the height of the cone is half the radius of the sphere, then the radius of the base of a cone which has the same volume as a sphere of $5$ cm radius, is
  1. $10$ cm
  2. $\displaystyle 10\sqrt{2}$ cm
  3. $\displaystyle \frac{5\sqrt{2}}{2}$ cm
  4. $\displaystyle \frac{10\sqrt{3}}{3}$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Volume of sphere = 4/3 * pi * 5^3 = 500/3 * pi. Volume of cone = 1/3 * pi * r^2 * h. Given h = 5/2. So 500/3 * pi = 1/3 * pi * r^2 * 5/2. 500 = r^2 * 5/2. r^2 = 200. r = sqrt(200) = 10*sqrt(2).

Multiple choice

A cylindrical container of radius 6 cm and height 15 cm is fulled with ice-cream. The whole ice-cream has to be distributed to 10 children in equal cones with hemispherical tops. If the height of the conical portion is four times the radius of its base, find the radius of the ice-cream cone

  1. 3 cm

  2. 1 cm

  3. 4 cm

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

The container volume is 540pi cubic centimetres, so each child's portion is 54pi cubic centimetres. A cone of radius r and height 4r plus a hemispherical top has volume 2pi r^3. Equating this to 54pi gives r^3 = 27, so r = 3 cm.

Multiple choice

A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is $10$ cm, and its base is of radius $3.5$ cm, find the volume of wood in the toy. $\displaystyle \left[ Use \,\,\, \pi = \frac{22}{7} \right]$
  1. $425$
  2. $290$
  3. $474.83$
  4. $205.33$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Volume of cylinder = pi * r^2 * h = (22/7) * 3.5^2 * 10 = 22 * 0.5 * 3.5 * 10 = 385. Volume of two hemispheres = 2 * (2/3) * pi * r^3 = (4/3) * (22/7) * 3.5^3 = (4/3) * (22/7) * 42.875 = 179.66. Volume of wood = 385 - 179.66 = 205.33.

Multiple choice

A right circular cone of volume $A$, a right circular cylinder of volume $M$, and a sphere of volume $C$ all have the same radius, and the common height of the cone and the cylinder is equal to the diameter of the sphere. Then
  1. $A - M + C = 0$
  2. $A + M = C$
  3. $2A = M + C$
  4. $A^{2} - M^{2} + C^{2} = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the radius be r and the height be h=2r. The volume of the cone A is (1/3)pi*r^2*(2r) = (2/3)pi*r^3. The volume of the cylinder M is pi*r^2*(2r) = 2pi*r^3. The volume of the sphere C is (4/3)pi*r^3. Substituting these, A - M + C = (2/3)pi*r^3 - 2pi*r^3 + (4/3)pi*r^3 = (6/3)pi*r^3 - 2pi*r^3 = 0.

Multiple choice

$16$ glass spheres each of radius $2$ cm are packed into a cuboidal box of internal dimensions 16 cm $\times$ 8 cm $\times$ 8 cm and then the box is filled with water. Find the volume of water filled in the box.
  1. $487.6$ $cm^3$
  2. $287.6$ $cm^3$
  3. $47.6$ $cm^3$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of the box is 16 * 8 * 8 = 1024 cm^3. The volume of 16 spheres with radius 2 cm is 16 * (4/3 * pi * 2^3) = 16 * (32/3 * pi) = 512/3 * pi, which is approximately 536.17 cm^3. Subtracting this from the box volume gives 1024 - 536.17 = 487.83 cm^3, which is closest to 487.6 cm^3.

Multiple choice

A solid is in the form of a cone mounted on a right circular cylinder both having same radii of their bases. Base of the cone is placed on the top base of the cylinder. If the radius of the base and height of the cone be 4 cm and 7 cm, respectively, and the height of the cylindrical part of the solid is 3.5 cm, the volume of the solid is equal to
  1. $\displaystyle 293\frac{1}{3}\, cm^{3}$
  2. $\displaystyle 293\frac{1}{5}\, cm^{3}$
  3. $\displaystyle 298\frac{1}{3}\, cm^{3}$
  4. $\displaystyle 298\frac{1}{5}\, cm^{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

A solid toy is in the form of a right circular with hemispherical ends. The total height of the solid is 19 cm and diameter of the cylinder is 7 cm. Find the volume and total surface area of the solid. (Take $\pi = \dfrac{22}{7}$)
  1. Volume = $681.66\, cm^{3};\,$ Total Surface Area = $ 478\, cm^{2}$
  2. Volume = $681.66\, cm^{3};\,$ Total Surface Area = $ 418\, cm^{2}$
  3. Volume = $641.66\, cm^{3};\,$ Total Surface Area = $ 478\, cm^{2}$
  4. Volume = $641.66\, cm^{3};\,$ Total Surface Area = $ 418\, cm^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The radius is 3.5 cm and the cylindrical portion has height 19 - 7 = 12 cm. Adding the cylinder volume and the volume of a sphere of radius 3.5 gives 641.66 cm^3, while the exposed area is 418 cm^2.

Multiple choice

A solid toy is in the form of a right circular cylinder with a hemispherical shape of one end and a cone at the other end. Then common diameter is $4.2 cm$ and the heights of the cyndrical and conical portions are $12 cm$ and $7 cm$, respectively. Find the volume of the solid. (Take $\pi$ = 22/7)
  1. $218.02\, cm^{3}$
  2. $178.66\, cm^{3}$
  3. $158.66\, cm^{3}$
  4. $138.66\, cm^{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The radius is 2.1 cm. Add the volumes of the cylinder, hemisphere, and cone: pi(2.1)^2(12) + (2/3)pi(2.1)^3 + (1/3)pi(2.1)^2(7), using pi = 22/7. This gives approximately 218.02 cm^3.

Multiple choice

Solid cylinders of equal volume are tightly packed in two layers in a rectangular box such that in each layer there are three rows of four such cylinders Find the percentage of volume of empty space in the box approximately.
  1. $19.27 \%$
  2. $20.54 \%$
  3. $21.43 \%$
  4. $22.36 \%$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The box contains 2 layers * 3 rows * 4 cylinders = 24 cylinders. Each cylinder has volume V = pi * r^2 * h. The box dimensions are 4 * (2r) by 3 * (2r) by 2 * h. Box volume = 8r * 6r * 2h = 96 * r^2 * h. Total cylinder volume = 24 * pi * r^2 * h. Empty space = 96 - 24 * pi. Percentage = (96 - 24 * 3.14159) / 96 * 100 = (96 - 75.398) / 96 * 100 = 21.46%.