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

Two circles touch each other externally. The sum of their area is $58$ $\pi\, cm^{2}$ and the distance between their center is $10\ cm$. Find the radii of the two circles.
  1. $7\ cm $and $3\ cm$
  2. $5\ cm $and $7\ cm$
  3. $3\ cm $and $2\ cm$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let radii be r1 and r2. r1 + r2 = 10 (distance between centers). Area sum: pi*r1^2 + pi*r2^2 = 58*pi => r1^2 + r2^2 = 58. (r1+r2)^2 - 2*r1*r2 = 58 => 100 - 2*r1*r2 = 58 => 2*r1*r2 = 42 => r1*r2 = 21. Roots of t^2 - 10t + 21 = 0 are 7 and 3.

Multiple choice

The sum of the radii of two circles is $84\ cm$ and the difference of their areas is $5544$ $cm^2$. Calculate the radii of the two circles.
  1. $52.5\ cm,\ 31.5\ cm$
  2. $52.5\ cm,\ 30.5\ cm$
  3. $50.5\ cm,\ 31.5\ cm$
  4. $50.5\ cm,\ 30.5\ cm$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

r1 + r2 = 84. pi * (r1^2 - r2^2) = 5544. pi * (r1 - r2)(r1 + r2) = 5544. (22/7) * (r1 - r2) * 84 = 5544. (r1 - r2) * 264 = 5544. r1 - r2 = 21. Solving r1+r2=84 and r1-r2=21: 2*r1 = 105 => r1 = 52.5. r2 = 31.5.

Multiple choice

A circular disc of area $\displaystyle A_{1}$ is given ;with its radius as the diameter a circular disc of area $\displaystyle A_{2}$ is cut out.If the area of the remaining disc is denoted by $\displaystyle A_{3}$ Then
  1. $\displaystyle A_{1}A_{3}<16 A_{2}^{2}$
  2. $\displaystyle A_{1}A_{3}>16 A_{2}^{2}$
  3. $\displaystyle A_{1}A_{3}=16 A_{2}^{2}$
  4. $\displaystyle A_{1}A_{3}>2 A_{2}^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A1 = pi*R^2. Radius of A2 is R/2, so A2 = pi*(R/2)^2 = pi*R^2/4 = A1/4. A3 = A1 - A2 = A1 - A1/4 = 3*A1/4. A1*A3 = A1*(3*A1/4) = 3*A1^2/4. 16*A2^2 = 16*(A1/4)^2 = 16*A1^2/16 = A1^2. Since 3/4 < 1, A1*A3 < 16*A2^2.

Multiple choice

The difference between the radii of the smaller circle and the bigger circle is $7 cm$ and the difference between the areas of the two circles is $1078$ sq cm. What is the radius of the smaller circle in cm?
  1. $28$
  2. $21$
  3. $17.5$
  4. $35$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let R and r be the radii. R - r = 7. pi * (R^2 - r^2) = 1078. pi * (R - r)(R + r) = 1078. (22/7) * 7 * (R + r) = 1078. R + r = 49. Solving R - r = 7 and R + r = 49 gives 2R = 56, R = 28, r = 21.

Multiple choice

The area of a circle drawn with its diameter as the diagonal of a cube of side of length 1 cm each is :
  1. $\displaystyle \frac{4\pi }{3}sq\, cm$
  2. $\displaystyle \frac{3\pi }{2}sq\,cm$
  3. $\displaystyle \frac{3\pi }{4}sq\,cm$
  4. $\displaystyle \frac{2\pi }{3}sq\, cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The diagonal of a cube with side 1 is sqrt(1^2 + 1^2 + 1^2) = sqrt(3). This is the diameter of the circle. Radius r = sqrt(3)/2. Area = pi * r^2 = pi * (sqrt(3)/2)^2 = pi * (3/4) = 3pi/4.

Multiple choice

Three circles of radii $ \displaystyle r_{1},r_{2}$ and $r_{3}$ are drawn concentric to each other. The radii $ \displaystyle r_{1}$ and $r_{2}$ are such that the area of the circle with radius $ \displaystyle r_{1}$ is equal to the area between the circles of radius $ \displaystyle r_{2}$ and $r_{1}$. The area between the circles of radii $ \displaystyle r_{3}$ and $r_{2}$ is equal to area between the circles of radii $ \displaystyle r_{2}$ and $r_{r}$. What is the value of $ \displaystyle r_{1}:r_{2}:r_{3}?$
  1. $ \displaystyle 1:\sqrt{2}:\sqrt{3}$
  2. $1:2:3$
  3. $1:\sqrt2:3$
  4. $1:2:\sqrt3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The first condition gives r2^2 - r1^2 = r1^2, so r2^2 = 2r1^2. The second condition gives r3^2 - r2^2 = r1^2, so r3^2 = 3r1^2. Therefore, the ratio is 1:sqrt(2):sqrt(3).

Multiple choice

The radius of the outer circumference of a cylindrical ring is $12 cm$ and the diameter of the cross-section is $1.5 cm$. Then, the volume of circular ring is
  1. $22.5\, cm^3$
  2. $125\, cm^3$
  3. $226\, cm^3$
  4. $1250\, cm^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The volume of a circular ring (torus) is given by the formula V = 2 * pi^2 * R * r^2, where R is the mean radius of the ring and r is the radius of the cross-section. Here, the cross-section radius is r = 1.5 / 2 = 0.75 cm, and the mean radius is R = 12 - 0.75 = 11.25 cm. Substituting these values gives V = 2 * pi^2 * 11.25 * 0.75^2, which is approximately 125 cubic cm.