Tag: perimeter, area and volume

Questions Related to perimeter, area and volume

The length of the base of a square pyramid is $2\ cm$ and the height is $6\ cm$. Calculate the volume.

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. $8\ cm^3$

  2. $6\ cm^3$

  3. $4\ cm^3$

  4. $2\ cm^3$

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Correct Option: A
Explanation:

Volume of square pyramid  $=\dfrac { 1 }{ 3 } \times { a }^{ 2 }\times h=\dfrac { 1 }{ 3 } \times 2\times 2\times 6=8{ cm }^{ 3 }$

The base of a right pyramid is an equilateral triangle of perimeter 8 cm 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. $160 cm^3$

  2. $1600 cm^3$

  3. $\dfrac {160}{3} cm^3$

  4. $\dfrac {5}{4} cm^3$

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Correct Option: A
Explanation:

Volume of right pyramid $=$ $\dfrac { 1 }{ 3 } \times area\quad of\quad base\times height\quad of\quad pyramid$

Now, base is equilateral $\triangle $, therefore,
area $=\dfrac { \sqrt { 3 }  }{ 4 } \times { \left( side \right)  }^{ 2 }$
Perimeter of triangle $=8cm$
$\therefore \quad \quad 3a=8\Rightarrow a=\dfrac { 8 }{ 3 } cm$
$\therefore \quad \quad area=\dfrac { \sqrt { 3 }  }{ 4 } \times \dfrac { 8 }{ 3 } \times \dfrac { 8 }{ 3 } =\dfrac { 16\sqrt { 3 }  }{ 9 } { cm }^{ 2 }$
Now,  Volume $=\dfrac { 1 }{ 3 } \times \dfrac { 16\sqrt { 3 }  }{ 9 } \times 30\sqrt { 3 } $
                        $=\dfrac { 160\times 3 }{ 9 } =\dfrac { 160 }{ 3 } { cm }^{ 3 }$

A right pyramid is on a regular hexagonal base. Each side of the base is $10$ m. Its height is $60$ m. 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. $5196 m^3$

  2. $5200 m^3$

  3. $5210 m^3$

  4. $5220 m^3$

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Correct Option: A
Explanation:
Volume of a hexagonal Pyramid $=$ $\displaystyle \frac{\sqrt{3}}{2}a^2h$
where $a$ $=$ side of the base. and $h$ $=$ height of the pyramid

$\therefore $Volume of a pyramid $=$ $\displaystyle \frac{\sqrt{3}}{2}{(10^2)}\times {60}$ $=$ $5196$ $m^3$

If a regular square pyramid has  a base of side 8 cm and height  of 30 cm, then its volume 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. 120 c.c.

  2. 240 c.c.

  3. 640 c.c.

  4. 900 c.c.

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Correct Option: C
Explanation:

Side of square base $=$ 8 cm.
Height of pyramid $=$ 30 cm
$\therefore$ Volume of square pyramid
$= \frac{1}{3}\times $area  of  base $\times$ height
$= \frac{1}{3}\times 82\times 30 = 640$ c.c.

If the volume of a prism is $1920$ $\sqrt{3} cm^3$ and the side of the equilateral base is $16$ $cm$, then the height (in cm) of the 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. $19$

  2. $20$

  3. $30$

  4. $40$

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Correct Option: C
Explanation:
Volume of prism $=$ Area of equilateral triangle $\times $ height
Now, Area of triangle $= \dfrac{\sqrt{3}}{4}a^2$
= $\dfrac{\sqrt{3}}{4} 16^2$ = $64 \sqrt{3}$
$1920 \sqrt{3} = 64\sqrt{3}\times $ height

$\therefore $ height $=$ $\displaystyle \dfrac{1920}{64}$
$= 30 \ cm$

The corner of a cube_has  been cut by the plane passing through mid-point of  the three edges meeting at that corner. If the edge of  the cube is of 2 cm length,  then the volume of the  pyramid thus cut off 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. $\dfrac{1}{24}cm^3$

  2. $\dfrac{1}{6}cm^3$

  3. $\dfrac{1}{48}cm^3$

  4. $6cm^3$

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Correct Option: B
Explanation:

The base of the pyramid thus out off will be a right angled triangle whose sides containing the right angle will be each equal to 1 cm. The height of the pyramid will also be equal to 1 cm. Hence, the volume will be equal to $\frac{1}{6}  cm^3$.

Each side of the base of a square pyramid is reduced by $20%$. By what percent must the height be increased so that the volume of the new pyramid is the same as the volume of the original 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. 20

  2. 40

  3. 46.875

  4. 56.25

  5. 71.875

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Correct Option: D
Explanation:
Let $a$ be the side of the square.
Length of side of square when reduced by $20\% = a-\dfrac{20a}{100}=0.8a$
Let $a _1=0.8a$
Volume of pyramid $V=\dfrac { 1 }{ 3 } \times $ Area of base $\times height=\dfrac{1}{3}A\times h$
Area of base with side $a = { a }^{ 2 }$ 
${ a }^{ 2 }=0.8a$

${V} _{ 1 }=\dfrac { 1 }{ 3 } \times { \left( { a } _{1} \right)  }^{ 2 }\times { h } _{ 1 }$ 
${V} _{ 1 }=V$ ....... [Given]

$\Rightarrow \dfrac { 1 }{ 3 } { a }^{ 2 }\times h=\dfrac { 1 }{ 3 } { \left( 0.8 \right)  }^{ 2 }{ a }^{ 2 }\times { h } _{ 1 }$

$\therefore { h } _{ 1 }=1.5625h$ 

$\implies \dfrac { { h } _{ 1 }-h }{ h } =1.5625-1=56.25%$

$\therefore$  'h' need to be increase by $56.25$