Chapter 12.5, Problem 15PPS

To determine

To find the volume of the given triangular pyramid.

Answer to Problem 15PPS

  $V=71.2$ cubic centimeters

Explanation of Solution

Given:

  

Height ( $h$ ) $=4.8$ centimeters

Right angled triangular base:

Hypotenuse $=10.2$ centimeters

Base $=5$ centimeters

Formula used:

Volume of pyramid, $V=\frac{1}{3}Bh$

where $B$ is the area of the base and $h$ is the height of the pyramid.

Pythagoras Theorem:

In right angle $\Delta ABC$

  $A{B}^2+B{C}^2=A{C}^2$

Area of a triangle $=\frac{1}{2}\times Base\times Height$

Calculation:

Since the given pyramid is a Right angled triangular pyramid, so it has a right angled triangular base.

  

In right angle $\Delta ABC$ ,

  $AB=$ height, $BC=$ base, and $AC=$ hypotenuse

So, by Pythagoras Theorem,

  $A{B}^2+B{C}^2=A{C}^2$

Now putting the values,

  $\Rightarrow A{B}^2+5^2={(10.2)}^2$

  $\Rightarrow A{B}^2={(10.2)}^2-5^2$

  $\Rightarrow A{B}^2=104.04-25$

  $\Rightarrow A{B}^2=79.04$

Now, by taking square root on both sides,

  $\Rightarrow AB=8.9$ centimeters

So, area of the base, $B=\frac{1}{2}\times Base\times Height$

  $\Rightarrow B=\frac{1}{2}\times 5\times 8.9$

  $\Rightarrow B=44.5$ square centimeters

Now for the volume ( $V$ ) of the pyramid,

  $\Rightarrow V=\frac{1}{3}Bh$

Now put the values of $B=44.5$ and $h=4.8$ ,

  $\Rightarrow \frac{1}{3}\times 44.5\times 4.8$

  $\Rightarrow \frac{213.6}{3}$

  $\Rightarrow V=71.2$ cubic centimeters

Conclusion:

Therefore, the volume of the given triangular pyramid is $71.2$ cubic centimeters.