Quantitative Aptitude

Mensuration of Solid Figures

147 Questions

Mensuration of solid figures deals with calculating the volume and surface area of 3D shapes. Common structures include cubes, cuboids, and rectangular prisms. These quantitative aptitude questions are standard in SSC, banking, and railway recruitment tests.

Cube volume formulasCuboid surface areaRectangular prismsDensity calculationsDimensional ratios

Mensuration of Solid Figures Questions

Multiple choice maths perimeter, area and volume surface area and volume of sphere surface area of a prism surface area of a prism and a pyramid

If a regular square pyramid has a base of side $8 cm$ and height of $30 cm$, then its volume is

  1. $120 cm^3.$
  2. $240 cm^3.$
  3. $640 cm^3.$
  4. $900 cm^3.$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Given that:
Side $a=8\ cm$, Height $h=30\ cm$
As we know that
Volume of regular square pyramid 
$\Rightarrow a^2\dfrac{h}{3}$
$\Rightarrow 8^2\dfrac{30}{3}$
$\Rightarrow 64\times 10$
$\Rightarrow 640\ cm^3$
This is the required solution.
Multiple choice maths spaces and boundaries - 2 area of rectangular paths comparing areas problems on areas

A closed box made of steel of uniform thickness has length breadth and height 12 dm, 10 sm and 8 dm respectively If the thickness of the steel sheet is 1 dm then the inner surface area is

  1. $ \displaystyle 456$ $\displaystyle dm^{2}$
  2. $ \displaystyle 376$ $\displaystyle dm^{2}$
  3. $ \displaystyle 264$ $\displaystyle dm^{2}$
  4. $ \displaystyle 696$ $\displaystyle dm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A closed box made by steel with uniform thickness length 12 dm ,breath  10 dm and 8 dm and thickness  of steel is 1 dm

Then inner length =12-2=10 dm ,breadth=10-2=8 dm and height=8-2=6 dm 
So surface area of box=$2(lw+wh+lh)=2(10\times 8)+(8\times 6)+(10\times 6)=2(80+48+60)=2\times 188=376 dm ^{2}$

Multiple choice maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

The side of a cube is equal to diameter of the sphere. The ratio of volumes 
of cube and sphere is

  1. $\frac{11}{12}$
  2. $\frac{22}{11}$
  3. $\frac{11}{21}$
  4. $\frac{21}{11}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the sides of cube be $s$ and radius of sphere be $r$

then $s=2r=(2r)^3=8r^3$
Volume of a cube$=s^3$
Volume of a sphere$=\cfrac{4}{3}\pi r^3$
Ratio of volume of cube and sphere$=\cfrac{8r^3}{\cfrac{4}{3}\pi r^3}$
$=\cfrac{8r^3}{\cfrac{4}{3}\times \cfrac{22}{7} r^3}\=\cfrac{21}{11}$

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

If a box is $\dfrac{1}{4}$ filled contains $5$ small cubes each of volume $1$ cubic units then find out the volume of the box.

  1. $25$ cu.
  2. $20$ cu.
  3. $15$ cu.
  4. $5$ cu.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since the box is $\dfrac {1}{4}$ filled with  the given cubes , we need to multiply the total volume of the given cubes by $4$ to get the total volume of thr box.

Volume of one cube is $1 cu.$
$\therefore $ Volume of 5 cubes will be $5\times 1 cu.=5cu.$
$\therefore $ Volume of the box will be $4\times 5 cu.=20cu.$

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

If the volumes of two cubes are in the ratio $8:1$, then the ratio of their edges is

  1. $8:1$
  2. $2\sqrt 2:1$
  3. $2:1$

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

Let $V _1$ and $V _2$ be two volume of cubes.

$l _1$ and $l _2$ be edges of the two cubes.
We know that,
Volume of cube $V=l^3$
So,
$\Rightarrow$  $\dfrac{V _1}{V _2}=\dfrac{l _1^3}{l _2^3}$

$\Rightarrow$  $\dfrac{8}{1}=\left(\dfrac{l _1}{l _2}\right)^3$             [ Given ]

$\therefore$  $\dfrac{l _1}{l _2}=\dfrac{2}{1}$

$\therefore$  Ratio of their edges is $2:1$.

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

The volume of a cube whose surface area is $96{cm}^{2}$, is

  1. $16\sqrt 2{cm}^{3}$
  2. $32{cm}^{3}$
  3. $64{cm}^{3}$
  4. $216{cm}^{3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $l$be the side of cube.

Surface area of cube $=6l^2$
$\Rightarrow$  $96=6l^2$                      [ Given ]
$\Rightarrow$  $l^2=16$
$\therefore$  $l=4\,cm$
Now,
$\Rightarrow$  Volume of cube $=l^3$
                                  $=(4)^3$
                                  $=64\,cm^3$

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

If each edge of a cube, of volume $V$, is doubled, then the volume of the new cube is

  1. $2V$
  2. $4V$
  3. $6V$
  4. $8V$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $a$ be the initial edge of the cube.

So, 
Volume of cube $V=a^3$
In the new cube,
Let $a'$ be the edge of new cube
$\therefore$  $a'=2a$               [ Given ]
Volume of new cube,
$V'=(a')^3$
      $=(2a)^3$
      $=8a^3$
      $=8V$                        [ Since, $a^3=V$ ]
Volume of the new cube is $8V.$

Multiple choice maths how big? how heavy? measuring volume volume of solids volume of cube and cuboid

If ${A} _{1},{A} _{2}$ and ${A} _{3}$ denote the areas of three adjacent faces of a cuboid, then its volume is

  1. ${A} _{1}{A} _{2}{A} _{3}$
  2. $2{A} _{1}{A} _{2}{A} _{3}$
  3. $\sqrt{{A} _{1}{A} _{2}{A} _{3}}$
  4. $\sqrt [ 3 ]{ { A } _{ 1 }{ A } _{ 2 }{ A } _{ 3 } } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

It is given that, $A _,A _2,A _3$ be the areas of $3$ adjacent faces of cuboid

Let $V$ be the volume of cuboid.
Let dimensions of cuboid $=l\times b\times h$
$A _1=l\times b$
$A _2=b\times h$
$A _3=h\times l$
$\Rightarrow$  $V=l\times b\times h$
Now,
$\Rightarrow$  $A _1A _2A _3=lb\times bh\times hl$
$\Rightarrow$  $A _1A _2A _3=l^2b^2d^2$

$\Rightarrow$  $A _1A _2A _3=(lbh)^2$
$\therefore$  $A _1A _2A _3=V^2$
$\therefore$  $V=\sqrt{A _1A _2A _3}$

Multiple choice maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

A regular hexagonal pyramid whose base perimeter is $60\ cm$ has an altitude of $30\ cm$, the volume of the pyramid (in cu. cm)is:

  1. $2958$
  2. $2598$
  3. $2859$
  4. $2589$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given : Perimeter of regular hexagon $=60\ cm$ and height $=30\ cm$

We know that, regular hexagon has all its sides of equal length
$\therefore$ Perimeter $=6a=60\ cm$,  where $a$ is side of hexagon
$\implies a=10\ cm$
There are exactly $6$ equilateral triangle
Area of one equilateral triangle $=\dfrac{\sqrt{3}}{4}a^2$
                                                     $=\dfrac{\sqrt{3}}{4}\times 10^2=25\sqrt{3}$
$\therefore$ Area of $6$ equilateral triangles $=6\times 25\sqrt{3}=150\sqrt{3}$
$\therefore\ Area\ of\ base = 150\sqrt{3}$
Volume $=\dfrac{1}{3}\times base \times height$
              $=\dfrac{1}{3}\times 150\sqrt{3}\times 30$
              $=1500\sqrt{3}=2598\ cu. m$
Hence, volume of pyramid is $2598\ cu. m$.

Multiple choice maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

A pyramid whose base is a regular pentagon of area $42\ {cm}^2$ and whose height is $7$ cm. What is the volume (in ${cm}^3$) of the pyramid?

  1. $98$
  2. $105$
  3. $126$
  4. $147$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given : Area of base of pyramid $=42\ cm^2$, height $=7\ cm$

We know that, Volume of pyramid $=\dfrac{1}{3}\times Area\ of\ base \times height$
                                                          $=\dfrac{1}{3}\times 42\times 7$
                                                          $=98\ cm^3$
Hence, volume of pyramid is $98\ cm^3$.