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

The surface areas of a solid sphere and a solid hemisphere are equal to S if their volumes are $\displaystyle V_{1}$ and $\displaystyle V_{2}$ respectively then $\displaystyle \frac{V_{1}}{V_{2}}$.
  1. $\displaystyle \sqrt{3}:2$
  2. $\displaystyle 3\sqrt{3}:8$
  3. $3 : 4$
  4. $\displaystyle 3\sqrt{3}:4$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a sphere of radius r1, S = 4 * pi * r1^2. For a hemisphere of radius r2, S = 3 * pi * r2^2. Equating them, 4 * pi * r1^2 = 3 * pi * r2^2, so r1^2 / r2^2 = 3 / 4, or r1 / r2 = sqrt(3) / 2. The volume ratio V1 / V2 = ((4/3) * pi * r1^3) / ((2/3) * pi * r2^3) = 2 * (r1 / r2)^3 = 2 * (sqrt(3)/2)^3 = 2 * (3 * sqrt(3) / 8) = 3 * sqrt(3) / 4.

Multiple choice

A toy is in the shape of a cone surmounting a hemisphere of radius 21 cm if the height of the cone is 28 cm, the total surface area of the toy is
  1. $\displaystyle 4824cm^{2}$
  2. $\displaystyle 4076cm^{2}$
  3. $\displaystyle 5082cm^{2}$

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

Radius r = 21, height h = 28. Slant height l = sqrt(21^2 + 28^2) = sqrt(441 + 784) = sqrt(1225) = 35. Surface area = pi * r * l (cone) + 2 * pi * r^2 (hemisphere). SA = (22/7) * 21 * 35 + 2 * (22/7) * 21 * 21 = 2310 + 2772 = 5082.

Multiple choice

The volume of material of a hollow sphere with external radius 10 cm and internal diameter 6 cm is nearly
  1. 4077 $\displaystyle cm^{3}$
  2. 4070 $\displaystyle cm^{3}$
  3. 4007 $\displaystyle cm^{3}$
  4. 4073 $\displaystyle cm^{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume of hollow sphere = (4/3) * pi * (R^3 - r^3). R = 10, r = 3 (radius is half of diameter 6). V = (4/3) * 3.14159 * (1000 - 27) = (4/3) * 3.14159 * 973 = 4077.15. This is approximately 4077.

Multiple choice

A solid sphere of radius $r$ is sliced by the planes passing through its centre and perpendicular to each other. The total surface area of each of the pieces so formed is
  1. $\displaystyle \frac{2}{3}\pi r^{2}$
  2. $\displaystyle \pi r^{2}$
  3. $\displaystyle \frac{4}{3}\pi r^{2}$
  4. $\displaystyle 2\pi r^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Slicing a sphere with two perpendicular planes through the center divides it into four equal parts. Each part has a surface area consisting of one-fourth of the sphere's surface (pi * r^2) plus two semicircular faces (each pi * r^2 / 2), totaling 2 * pi * r^2.

Multiple choice

The volume of a solid in the form of a cylinder with hemispherical ends whose extreme length is 23 cm and radius 3 cm is
  1. 594 $ \displaystyle cm^{3} $
  2. 694 $ \displaystyle cm^{3} $
  3. 794 $ \displaystyle cm^{3} $
  4. 894 $ \displaystyle cm^{3} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The solid consists of a cylinder of length (23 - 3 - 3) = 17 cm and two hemispheres of radius 3 cm. Volume = pi * r^2 * h_cyl + (4/3) * pi * r^3 = pi * 3^2 * 17 + (4/3) * pi * 3^3 = 153 * pi + 36 * pi = 189 * pi. Using pi approx 3.14, 189 * 3.14 = 593.46, which is approximately 594.

Multiple choice

The longest needle that can be placed in a cylinder of radius r units and height h units is _____ (in units)
  1. $\displaystyle \sqrt{4r^{2}+h^{2}}$
  2. $\displaystyle \sqrt{3r^{2}+h^{2}}$
  3. $\displaystyle \sqrt{4r^{2}-h^{2}}$
  4. $\displaystyle \sqrt{3r^{2}+2h^{2}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The longest needle in a cylinder is the space diagonal of the cylinder, which connects opposite points on the circular bases. By the Pythagorean theorem, the length is sqrt((2r)^2 + h^2) = sqrt(4r^2 + h^2).