Tag: volume of prism and pyramid

Questions Related to volume of prism and pyramid

A frustum of a pyramid has an upper base $100\ m$ by $10\ m$ and a lower base of $80\ m$ by $8\ m$. if the altitude of the frustum is $5\ m$, find its volume (in cu. m).

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

  1. $4567.67$

  2. $3873.33$

  3. $4066.67$

  4. $2345.98$

Show answer
Correct Option: C
Explanation:

Give : Height of pyramid$(h)=5\ m$ 

It is a pyramid with rectangular base.
Edge of upper base $length(L)=100\ m, breadth(B)=10\ m$

Edge of lower base $length(l)=80\ m, breadth(b)=8\ m$
Area of upper base$(A _{1})=L\times B=100\times 10 = 1000\ m^2$
Area of lower base$A _{2}=l\times b=640\ m^2$
Volume of frustum$(V)=\dfrac{h}{3}(A _{1} + A _2 + \sqrt{A _1 \times A _2})$
$\implies$$(V)=\dfrac{5}{3}(1000 + 640 + \sqrt{1000 \times 640})$
                 $=\dfrac{5}{3}(1640+800)=\dfrac{5}{3}(2440)=4066.666667$
Hence, volume$(V)=4066.67\ cu. m$

A regular triangular pyramid has an altitude of $9\ m$ and a volume of $187.06\ cu.\ m$. What is the base edge in meters?

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

  1. $12$

  2. $13$

  3. $14$

  4. $15$

Show answer
Correct Option: A
Explanation:

Given : Altitude$(height\quad (h))=9\ m$, volume $=187.06\ cu.\ m$

We know that, Volume $=\dfrac{1}{3}Bh$, where $B=x^2 \sin \theta$
$\implies 187.06=\dfrac{1}{3} \left(\dfrac{1}{2}x^2 (\sin \theta)\right) (9)$,  where $x$ is base edge
$\implies 187.06=\dfrac{1}{3} \left(\dfrac{1}{2}x^2 \sin60\right)9$
$\implies x^2=143.9988=144$
$\therefore\ x=12\ m$

The frustum of a regular triangular pyramid has equilateral triangles for its bases. The lower and upper base edges are $9\ m$ and $3\ m$, respectively. If the volume is $118.2\ cu.\ m$, how far apart (m) are the base?

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

  1. $9$

  2. $8$

  3. $7$

  4. $10$

Show answer
Correct Option: C
Explanation:

Given : Volume $=118.2\ cu. m$

Upper base edge $=9\ m$, lower base edge $=3\ m$
We know that, 
Volume $=\dfrac{h}{3}(A _{1}+A _{2}+\sqrt{A _{1} A _{2}})$ ....... $(1)$, where $A _{1}, A _{2}$ are area of upper and lower bases.
$A _{1}=\dfrac{\sqrt{3}}{4}\times 9^2=35.074$
$A _{2}=\dfrac{\sqrt{3}}{4}\times 3^2=3.897$
From $(1)$ we get,
$118.2 = \dfrac{h}{3}(35.074+3.897+\sqrt{35.074\times 3.897})$
$\implies 118.2=\dfrac{h}{3}(38.971+11.6911)$
$\implies 118.2\times 3=h(50.6621)$
$\implies h=7\ m$
Hence, the bases are $7\ m$ far from each other.

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:

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

  1. $2958$

  2. $2598$

  3. $2859$

  4. $2589$

Show answer
Correct Option: B
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 $2598\ cu. m$.

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?

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

  1. $98$

  2. $105$

  3. $126$

  4. $147$

Show answer
Correct Option: A
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 $98\ cm^3$.

General formula of volume of a prism is:

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

  1. Area of base $\times$ height

  2. Area of triangle $\times$ height

  3. Area of square $\times$ height

  4. Area of rectangle $\times$ height

Show answer
Correct Option: A
Explanation:

The volume of a general 3d figure is simply = Area of  Base $\times$ Its height.

This applies for prism also.

If base and height of a prism and pyramid are same, then the volume of a pyramid is:

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

  1. $\dfrac{1}{3}\times$ Volume of prism

  2. ${3}\ \times$ Volume of prism

  3. $\dfrac{1}{2}\times$ Volume of prism

  4. $2\ \times$ Volume of prism

Show answer
Correct Option: A
Explanation:

If a pyramid and a prism have the same base and height, their volumes are always in the ratio of $1:3\times $Volume of prism.  

General formula to find volume of a pyramid is:

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

  1. $\dfrac{\text{Base Area} \times \text{Height}}{2}$

  2. $2(\text{Base Area} \times \text{Height})$

  3. $\dfrac{\text{Base Area} \times \text{Height}}{3}$

  4. $3(\text{Base Area} \times \text{Height})$

Show answer
Correct Option: C
Explanation:

Pyramid is a structure whose outer surfaces are triangular and converge to a 

single point at the top.

Its base is a polygon i.e triangle, rectangle, pentagon etc.

So, volume is found by finding area of its base times height 

$\therefore$ Volume of a pyramid $=\dfrac{\text{Base Area} \times \text{Height}}{3}$

The base of the right pyramid is a square of side 16 cm and height 15 cm. Its volume $(cm^{3})$ will be

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

  1. $3840$

  2. $1920$

  3. $1280$

  4. $960$

Show answer
Correct Option: C
Explanation:

Area of the base $=$ $(16 \times 16) cm^2$
Volume of the pyramid $=$ $\frac{1}{3} \times B h$
Volume of the pyramid $=$ $\frac{1}{3}\left ( 16\times 16 \right )\times 15$
$= 1280 cm^2$

The base of a right pyramid is an equilateral triangle of perimeter $8$ dm and the height of the pyramid is $30$$\sqrt{3}$ cm. The volume of the pyramid is

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

  1. $1600$ cm$^{3}$

  2. $16000$ cm$^3$

  3. $\displaystyle \frac{16000}{3} cm^3$

  4. $\displaystyle \frac{5}{4} cm^3$

Show answer
Correct Option: B
Explanation:
Base of pyramid is an equilateral triangle of parameter
$8dm=80cm$
Let the side of the equilateral triangle be $'a'cm$
$\therefore $Parameter of equilateral triangle$=3a$
$\Rightarrow 3a=80\Rightarrow a=\cfrac { 80 }{ 3 } cm$
Height of pyramid$=30\sqrt { 3 } cm$
Volume of pyramid=Area of base $\times $ height
$=\cfrac { \sqrt { 3 }  }{ 4 } { a }^{ 2 }\times 30\sqrt { 3 } $

$=\cfrac { \sqrt { 3 }  }{ 4 } \times \cfrac { 80 }{ 3 } \times \cfrac { 80 }{ 3 } \times 30\sqrt { 3 } $

$=\cfrac { 3 }{ 4 } \times \cfrac { 80 }{ 3 } \times 80 \times 10 $

$=\cfrac { 1 }{ 4 } \times 80 \times 80 \times 10 $

$=20 \times 80 \times 10 $

$=16000cm^3$