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 right circular cone and a sphere have equal volumes. If the radius of the base of the cone is $2x$ and the radius of the sphere is $3x$, find the height of the cone in terms of $x$.
  1. $x$
  2. $\dfrac{3x}{2}$
  3. $\dfrac{4x}{3}$
  4. $20x$
  5. $27x$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Volume of cone = (1/3) * pi * (2x)^2 * h = (4/3) * pi * x^2 * h. Volume of sphere = (4/3) * pi * (3x)^3 = (4/3) * pi * 27x^3. Equating: (4/3) * pi * x^2 * h = (4/3) * pi * 27x^3. h = 27x.

Multiple choice

A solid is in the shape of a cylinder surmounted on a hemisphere. If the diameter and the total height of the solid are $21cm,25.5cm$ respectively, then find its volume.
  1. $7623{cm}^{3}$
  2. $723{cm}^{3}$
  3. $763{cm}^{3}$

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

The solid consists of a hemisphere (radius 10.5 cm) and a cylinder (radius 10.5 cm, height 25.5 - 10.5 = 15 cm). Volume = (2/3) * pi * r^3 + pi * r^2 * h. Using pi = 22/7, Volume = (2/3)(22/7)(10.5)^3 + (22/7)*(10.5)^2 * 15 = 2425.5 + 5197.5 = 7623 cm^3.

Multiple choice

Choose the correct answers from the alternatives given. If a cone, a hemisphere and a cylinder stand on the same base and have height equal to the radius of the base, find out the ratio of their volumes.

  1. 1:3:2

  2. 2:3:1

  3. 3:2:1

  4. 1:2:3
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let radius = r, height = r. Volume of cone = (1/3)pi*r^2*h = (1/3)pi*r^3. Volume of hemisphere = (2/3)pi*r^3. Volume of cylinder = pi*r^2*h = pi*r^3. Ratio = (1/3) : (2/3) : 1 = 1 : 2 : 3.

Multiple choice

There are two vessels - one is in the shape of a cylinder and the other in the shape of a right circular cone. Both the vessels have the same height and the same base radius. The cylindrical vessel and the conical vessel are filled with milk and water respectively and are both filled to half of their maximum heights. The cone is standing on its vertex. The contents of the conical vessel are emptied into the cylindrical vessel. What is the ratio of water to milk in the cylindrical vessel now
  1. $1 : 1$
  2. $1: 4$
  3. $1 : 9$
  4. $1 : 12$
Reveal answer Fill a bubble to check yourself
D Correct answer
Multiple choice

The radius of the base of the height of a cone are $3\,cm$ and $5\,cm$ respectively whereas the radius of the base and height of a cylinder are $2\,cm$ and $4\,cm$ respectively. The ratio of the volume of cone to that of the cylinder is :
  1. $1:3$
  2. $15:8$
  3. $15:16$
  4. $45:16$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Volume of cone = (1/3) * pi * r^2 * h = (1/3) * pi * 3^2 * 5 = 15 * pi. Volume of cylinder = pi * r^2 * h = pi * 2^2 * 4 = 16 * pi. Ratio = 15/16.

Multiple choice

A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank.
  1. $ 1.063487 $
  2. $ 0.63487 $
  3. $ 0.063487 $

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

Volume of hemisphere = (2/3) * pi * r^3. Outer radius R = 1m + 0.01m = 1.01m. Inner radius r = 1m. Volume of iron = (2/3) * pi * (1.01^3 - 1^3) = (2/3) * 3.14159 * (1.030301 - 1) = (2/3) * 3.14159 * 0.030301 = 0.063487 cubic meters.

Multiple choice

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom.Find the volume of water left in the cylinder, if the radius is 60 cm and its height is 180 cm.
  1. $1133400 cm^3$
  2. $1130400 cm^3$
  3. $1135400 cm^3$
  4. $1138400 cm^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The volume of the solid is the sum of the cone (1/3 * pi * r^2 * h) and the hemisphere (2/3 * pi * r^3). With r=60 and h=120, the solid volume is 1/3 * pi * 60^2 * 120 + 2/3 * pi * 60^3 = 144000*pi + 144000*pi = 288000*pi. The cylinder volume is pi * r^2 * H = pi * 60^2 * 180 = 648000*pi. The remaining water is (648000 - 288000) * pi = 360000 * 3.14 = 1130400.

Multiple choice

A solid is in the form of a right circular cone mounted on a solid hemisphere with same radius is made from a piece of metal. The radius of the hemisphere is $\displaystyle \frac{1}{3}$ of the vertical height of the conical part. If the radius of the base of the cone is r, the volume of the piece of metal is
  1. $\displaystyle \frac{11}{3}\pi\, r^{3}$
  2. $\displaystyle \frac{8}{3}\pi\, r^{3}$
  3. $\displaystyle \frac{7}{3}\pi\, r^{3}$
  4. $\displaystyle \frac{5}{3}\pi\, r^{3}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Radius of hemisphere = r. Height of cone = 3r. Volume = (2/3) * pi * r^3 (hemisphere) + (1/3) * pi * r^2 * (3r) (cone) = (2/3) * pi * r^3 + pi * r^3 = (5/3) * pi * r^3.

Multiple choice

From a solid right circular cylinder with height $h$ and radius of the base $r$, a right circular cone of the same height and same base is removed. Find the volume of the remaining solid.
  1. $\displaystyle \frac{2}{3}\, \pi r^{2}h$
  2. $ \pi r^{2}h$
  3. $2 \pi r^{2}h$
  4. $\displaystyle \frac{1}{3}\, \pi r^{2}h$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume of cylinder = pi * r^2 * h. Volume of cone = (1/3) * pi * r^2 * h. Remaining volume = pi * r^2 * h - (1/3) * pi * r^2 * h = (2/3) * pi * r^2 * h.

Multiple choice

A solid is in the form of a cone of vertical height $9 cm$ mounted on the top base of a right circular cylinder of height $40 cm$. The radius of the base of the cone and that of the cylinder are both equal to $7 cm$. Find the weight of the solid if $1\, cm^{3}$ of the solid weighs $4 gm.$
  1. $26.488 kg$
  2. $25.488 kg$
  3. $24.488 kg$
  4. $23.488 kg$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume of cylinder = pi * r^2 * h = (22/7) * 7 * 7 * 40 = 6160 cm^3. Volume of cone = (1/3) * pi * r^2 * h = (1/3) * (22/7) * 7 * 7 * 9 = 462 cm^3. Total volume = 6622 cm^3. Weight = 6622 * 4 = 26488 gm = 26.488 kg.