Mathematics · Quantitative Aptitude

Mensuration of Solids

194 Questions

Mensuration of solids focuses on calculating the volume and surface area of three dimensional shapes like cylinders and cones. These geometry problems require applying standard mathematical formulas. They frequently appear in state public service and engineering exams.

Cylinder volume and areaCone slant heightSurface area ratiosHollow pipe calculationsDimensional optimization

Mensuration of Solids Questions

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A closed cone tank radius $7$ cm and height $10$ cm is made from a sheet of aluminium. How much sheet is required?

  1. $144cm^2$
  2. $\displaystyle 22\sqrt { 149 } cm^2$
  3. $\displaystyle (22\sqrt { 149 } +144) cm^2$

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

Total surface area of cone  = Area of base + Area of curved surface
$\displaystyle =\quad \pi { r }^{ 2 }+\pi rs$

$\displaystyle =\frac { 22 }{ 7 } \times 7\times 7+\frac { 22 }{ 7 } \times 7\left( \sqrt { 149 }  \right) $
$\displaystyle s=\sqrt { { h }^{ 2 }+{ r }^{ 2 } } $
$\displaystyle =\sqrt { 149+100 } $
$\displaystyle =144+22\sqrt { 149 } $

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Find  the radius of the base of a right circular cone which has a lateral surface area of $6\pi$ and a slant height of $6$ ( in standard units )

  1. $0.50$
  2. $0.75$
  3. $1.00$
  4. $1.25$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, Lateral surface area $=6 \pi$ and slant height $=6$

Let the radius of base be $r$ and slant height of cone be $l = 6$.
Lateral surface area is equal to $\pi rl = \pi \times r \times 6 = 6\pi$
$\Rightarrow 6 \pi= \pi \times r \times 6$
$\Rightarrow r=1$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Find the slant height and vertical height of a Cone with radius $5.6$ cm and curved surface area $158.4$ cm$^2$.

  1. $8.07$
  2. $7.05$
  3. $8$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Radius $=5.6$ cm, vertical height $= h$, slant height $=$ $l$
Curved Surface Area of cone $=\pi r l = 158.4 cm^2$
$\Rightarrow \dfrac{22}{7} \times 5.6 \times l = 158.4$
$\Rightarrow l = \dfrac{158.4 \times 7}{22 \times 5.6} = \dfrac{18}{2} = 9$ cm
We know $l^2 = r^2 + h^2$
Thus $h^2 = l^2 - r^2 $

$= 9^2 - (5.6)^2$
$= 81 - 31.36$
$= 49.64$
$h = \sqrt{49.64}$
$h = 7.05 $ cm (approx.)

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The curved surface area of right circular cone with height $24$ m and radius $7$ m is

  1. $500\ \text{m}^{2}$
  2. $550\ \text{m}^{ 2 }$
  3. $607\ \text{m}^{ 2 }$
  4. $650\ \text{m}^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Height of cone$=24m$
Radius of cone$=7m$
Slant height of cone$=\sqrt { { 24 }^{ 2 }+{ 7 }^{ 2 } } =\sqrt { 576+49 } =\sqrt { 625 } =25m$
CSA of cone$=\pi rl=\cfrac { 22 }{ 7 } \times 7\times 25=550 m^2$
Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A cone and a cylinder have the same base area. They also have the same curved surface area. If the height of the cylinder is $3$ m, then the slant height of the cone (in m) is

  1. $3$
  2. $4$
  3. $6$
  4. $7$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given the radius of cylinder and cone are the same because there base areas are same.
Curved surface of cylinder $=$ curved surface area of the cone
$\therefore 2 \pi r h = \pi r l$
$\therefore l = 2h $
Given, height of cylinder $= 3$ cm
Therefore, slant height of cone $= 6$ cm

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The curved surface area of a right circular cone with height $24$ cm and radius $7$ cm is

  1. $500 cm^2$
  2. $550 cm^2$
  3. $607 cm^2$
  4. $650 cm^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given,
Radius of cone $= 7$ cm
Height of cone $= 24$ cm
Curved surface area = $\pi r \sqrt{(r^2 + l^2)}$
= $\pi \times 7 (\sqrt{(7^2 + (24)^2)}$
= $\pi \times 70\times 25$
= $550 cm^2$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Consider two cones, the curved surface area of one being twice that of the other and the slant height of the later being twice that of the former. The ratio of the radius of the later cone to that of the former is

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

Let the curved surface areas of two cones be $C _1$ and $C _2$

And their slant heights be $l _1$ and $l _2$
Given, $C _1=2C _2$
$l _2=2l _1$

$\therefore \pi r _1l _1=2\pi r _2.2l _1$
$\therefore r _1=4r _2$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The curved surface area of a right cone is $\displaystyle 286 m^{2}$ and its the slant height is 13 m, then volume is

  1. $\displaystyle 821.389m^{3}$
  2. $\displaystyle 852.258m^{3}$
  3. $\displaystyle 364.369m^{3}$
  4. $\displaystyle 281.164m^{3}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Curved surface area of a cone $= \pi rl$  where r is
the radius of the cone and l is the slant height.
Hence, CSA of this cone, $ = \frac {22}{7} \times r \times 13 = 286 $
$ => r = 7  m $

For a cone, l $ = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $ where lis the slant height.

Hence, $ 13 = \sqrt { { h }^{ 2 }+  {7}^{ 2 } } $ 

$ 169 = { h }^{ 2 } + 49 $

$ { h }^{ 2 } = 120 $

$ h = 2 \sqrt {30}  m $

Hence, volume of this cone $ = \frac { 1 }{ 3 } \times \frac { 22 }{ 7 } \times { 7 }^{ 2 }\times 2 \sqrt {30} = \frac {154}{3} \sqrt {30} { m }^{ 3 } $


Multiple choice chemistry substances in the surroundings - their states and properties measurement of density properties of substances fundamental and derived units

If the length of the cylinder is measured as 25 mm, the diameter is 3.09cm and mass of the cylinder is measured as 50.0 gm find the density of the cylinder in proper significant figures.(in $gm/cm^3$)

  1. 2.700 $gm/cm^3$
  2. 2.7 $gm/cm^3$
  3. 0.27 $gm/cm^3$
  4. 2.70$gm/cm^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$p\, =\, \displaystyle \frac{m}{\pi (d^2\, /\, 4)h}$
$p\, =\, \displaystyle \frac{(50.0) gm}{3.14\, \times\, (3.09/2)^2\, \times\, (25\, \times\, 10^{-1}) cm^3}\, =\, 2.7\, gm/cm^3$
(in two S.F.)

Multiple choice physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

The radius and height of a cone are measured as $6cms$ each by scale in which there is an error of $0.01cm$ in each cm. then the approximate error in its volume is.

  1. $.14$
  2. $.12$
  3. $.36$
  4. $0.16$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Formula,

$V=\pi r^2 \dfrac{h}{3}$

$=\pi\times 6^2 \dfrac{6}{3}=226.19$

$\dfrac{\Delta V}{V}=\dfrac{\pi}{3}6 \times 0.01 \times 0.01=0.000628$

The change in volume is,

$V=0.000628\times 226.19=0.14$%

Multiple choice physics units and measurement: error analysis significant figures significant figures and rounding of digits units and measurements

The diameter and height of a cylinder are measured by a meter scale to be $12.6\pm 0.1$ cm and $34.2\pm 0.1$ cm ,respectively. What will be the value of its volume in appropriate significant figures?

  1. $4264\pm 81{ cm }^{ 3 }$
  2. $4260\pm 80{ cm }^{ 3 }$
  3. $4264.4\pm 81.0{ cm }^{ 3 }$
  4. $4300\pm 80{ cm }^{ 3 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Volume V = pi * r^2 * h. With diameter d = 12.6, r = 6.3. V = pi * (6.3)^2 * 34.2 = 4264.4. The uncertainty is calculated via relative errors: deltaV/V = 2(deltaD/D) + deltaH/H = 2(0.1/12.6) + 0.1/34.2 = 0.0158 + 0.0029 = 0.0187. deltaV = 4264.4 * 0.0187 = 79.7, which rounds to 81.

Multiple choice physics units and measurement: error analysis significant figures significant figures and rounding of digits units and measurements

The diameter and height of a cylinder aremeasured by a meter scale to be 1.6$\pm 0.1$$\mathrm { cm }$ and 5.0$\pm 0.1 \mathrm { cm }$ , respectively. What will be the value of its volume inappropriate significant figures? 

  1. 4300$\pm 80 \mathrm { cm } ^ { 3 }$
  2. 4260$\pm 80 \mathrm { cm } ^ { 3 }$
  3. 4264$\pm 81 \mathrm { cm } ^ { 3 }$
  4. 10.0$\pm .325 \mathrm { cm } ^ { 3 }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The calculation for volume V = pi * r^2 * h with r = 0.8 and h = 5.0 yields approximately 10.05. The uncertainty calculation matches option D.

Multiple choice

What is the name of the theorem that states that the surface area of a cone is equal to the sum of the areas of its base and its lateral surface?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Angle Sum Theorem

  4. Surface Area of a Cone Formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Surface Area of a Cone Formula states that the surface area of a cone is equal to the sum of the areas of its base and its lateral surface. This formula is attributed to Indian mathematicians.