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 sphere of radius $r$ lies inside a cube and touches each of the six sides of the cube. Calculate the volume of the cube in terms of $r$.
  1. $\displaystyle { r }^{ 3 }$
  2. $\displaystyle 2{ r }^{ 3 }$
  3. $\displaystyle 4{ r }^{ 3 }$
  4. $\displaystyle \frac { 4 }{ 3 } \pi { r }^{ 3 }$
  5. $\displaystyle 8{ r }^{ 3 }$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

If a sphere of radius r is inscribed in a cube, the side length of the cube is equal to the diameter of the sphere, which is 2r. The volume of the cube is (side)^3 = (2r)^3 = 8r^3.

Multiple choice

The largest possible sphere is carried out a wooden solid cube of side $7$ cm. Find the volume of the wood left. $\left[\text{Use } \displaystyle \pi =\dfrac { 22 }{ 7 }\right]$
  1. $163.3$
  2. $164$
  3. $165$
  4. $170$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of the cube is 7^3 = 343. The largest sphere inside has a diameter of 7, so radius = 3.5. Volume of sphere = (4/3) * pi * r^3 = (4/3) * (22/7) * (3.5)^3 = 179.66. Wood left = 343 - 179.66 = 163.33.

Multiple choice

A rectangular sheet of dimensions $1.57$ m $\times 4.16$ m was brought to form an ice-cream cone of radius $2.5$ cm and height $6 $ cm. Then find out how many ice-cream cones can be prepared from that ?
  1. $320$
  2. $1280$
  3. $640$
  4. $512$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area of sheet = 1.57 * 4.16 = 6.5312 m^2 = 65312 cm^2. Surface area of cone (lateral) = pi * r * l. l = sqrt(r^2 + h^2) = sqrt(2.5^2 + 6^2) = sqrt(6.25 + 36) = 6.5. Area = 3.14 * 2.5 * 6.5 = 51.025 cm^2. Number of cones = 65312 / 51.025 = 1280.

Multiple choice

A solid metal sphere of surface area $S_1$ is melted and recast into a number of smaller spheres. $S_2$ is the sum of the surface areas of all the smaller spheres. Then.
  1. $S_1 > S_2$
  2. $S_2 > S_1$
  3. $S_1 = S_2$
  4. $S_1 = S_2$ only if all the smaller spheres of equal radii
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Volume is conserved, but surface area increases when a large sphere is broken into smaller ones. For a fixed volume, the sphere has the minimum surface area. Thus, S2 > S1.

Multiple choice

The surface area of cylindrical pipe, open at both ends is 628 sq.m. The difference between its radius and length is 15 m, the length being larger. If the pipe was closed at one end, the amount of water that it can hold is
  1. $500 \pi cu.m$
  2. $525 \pi cu.m$
  3. $550 \pi cu.m$

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

Surface area of open cylinder = 2 * pi * r * h = 628. Given h - r = 15, so h = r + 15. 2 * pi * r * (r + 15) = 628. Using pi = 3.14, 2 * 3.14 * r * (r + 15) = 628, so r(r + 15) = 100. r^2 + 15r - 100 = 0. (r + 20)(r - 5) = 0, so r = 5 and h = 20. Volume = pi * r^2 * h = pi * 25 * 20 = 500 * pi.

Multiple choice

Three congruent circles each of radius $4$ cm touches each other. What is the area (in $cm^{2}$) of the portion included between them?
  1. $8\pi$
  2. $16\sqrt {3} - 8\pi$
  3. $16\sqrt {3} - 4\pi$
  4. $16\sqrt {3} - 2\pi$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The centers of the three mutually touching circles form an equilateral triangle with side length equal to twice the radius, which is 8 cm. The area of this equilateral triangle is (sqrt(3)/4) * 8^2 = 16 * sqrt(3) square centimeters. The area of the three 60-degree sectors inside the triangle is 3 * (60/360) * pi * 4^2 = 8 * pi square centimeters, so the enclosed area is 16 * sqrt(3) - 8 * pi.

Multiple choice

$OABC$ is a rhombus whose three vertices $A, B$ and $C$ lie on a circle with centre $O$. If the radius of the circle is $10$ cm. Find the area of the rhombus.
  1. $25$ $\sqrt{3}$ sq. cm.
  2. $50$ $\sqrt{3}$ sq. cm.
  3. $75$ $\sqrt{3}$ sq. cm.
  4. $20$ $\sqrt{3}$ sq. cm.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a rhombus OABC with O as the center, the sides OA, OB, OC are radii (10 cm). The triangles OAB and OCB are equilateral triangles. The area is 2 * (sqrt(3)/4 * 10^2) = 50 * sqrt(3).