Question

How do you find the volume of a square pyramid with height $ 12\,cm $ and slant height $ 15\,cm $ ?

Answer 1

Hint: In order to determine the volume of the square pyramid, we will determine the value of the side of the square from the formula of slant height of the square pyramid. Then, we will substitute the values in the formula of volume of the square pyramid and evaluate it to determine the volume of the square pyramid.

Complete step-by-step answer:

Here, we need to determine the volume of a square pyramid height $ 12\,cm $ and slant height $ 15\,cm $ .
The volume of the square pyramid is $ \dfrac{1}{3}{a^2}h $ .
Where $ a $ is the base length of the square base of the square pyramid
And, $ h $ is the height of the square pyramid.
The slant height of the square pyramid is given by,

By Pythagorean Theorem, we know that $ {s^2} = {r^2} + {h^2} $
Since $ r = \dfrac{a}{2} $ ,
 $\Rightarrow {s^2} = {\left( {\dfrac{a}{2}} \right)^2} + {h^2} $
 $ {s^2} = \left( {\dfrac{1}{4}} \right){a^2} + {h^2} $
Now, let us substitute the values of $ h $ and $ s $ , we have,
 $\Rightarrow {\left( {15} \right)^2} = \left( {\dfrac{1}{4}} \right){a^2} + {\left( {12} \right)^2} $
 $\Rightarrow 225 = \left( {\dfrac{1}{4}} \right){a^2} + 144 $
 $\Rightarrow \left( {\dfrac{1}{4}} \right){a^2} = 225 - 144 $
 $\Rightarrow \left( {\dfrac{1}{4}} \right){a^2} = 81 $
 $\Rightarrow {a^2} = 81 \times 4 $
 $\Rightarrow {a^2} = 324 $
 $\Rightarrow a = \sqrt {324} $
Therefore, $ a = 18\,cm $
Now, let us substitute the values in the volume of the square pyramid,
 $\Rightarrow V = \dfrac{1}{3}{a^2}h $
 $\Rightarrow V = \dfrac{1}{3} \times {\left( {18} \right)^2} \times 12 $
 $\Rightarrow V = 1296\,c{m^3} $
Hence, the volume of the square pyramid is $ 1296\,c{m^3} $ .
So, the correct answer is “ $ 1296\,c{m^3} $ ”.

Note: A square pyramid is a type of pyramid with a square- shaped base and 4 triangular faces which meet each other at a vertex.
Now, let us know the important formulas of a square pyramid,
Volume of the square pyramid, $ V = \dfrac{1}{3}{a^2}h $
Lateral surface of the square pyramid, $ F = a\sqrt {4{h^2} + {a^2}} $
Surface area of the square pyramid, $ S = F + {a^2} $
Height of side face of the square pyramid, $ b = \sqrt {{h^2} + {{\left( {\dfrac{a}{2}} \right)}^2}} $
Angle of inclination of the square pyramid, $ \phi = {\tan ^{ - 1}}\left( {\dfrac{{2h}}{a}} \right) $ .
Whenever we are facing these types of questions, we need to be good at the formulas like volume, total surface area, lateral surface area, slant height of the cube, cuboids, cylinder, etc.