Chapter 12.2, Problem 31WE

To determine

To find: The volume of the give regular square pyramid.

Answer to Problem 31WE

The volume of the regular square pyramid is $246\text{cubic}\text{unit}$ .

Explanation of Solution

Given information:

Consider the given shape of square pyramid.

  

Calculation:

Use trigonometry to determine the vertical and horizontal components. Let the height of the pyramid is $h$ and the base edge is $a$ .

  

Calculate the height of the pyramid using vertical component.

  $h=10\sin 42°$

Since the base of the pyramid is a square. So, each corner angle will be a right angle. Apply trigonometric identity in $\Delta \text{ABC}$ to calculate the base edge.

  $\begin{array}{c}\cos 45°=\frac{10\cos 42°}{a}\\ a=\frac{10\cos 42°}{\cos 45°}\\ a=\frac{10\cos 42°}{\left(\frac{1}{\sqrt{2}}\right)}\left\{\cos 45°=\frac{1}{\sqrt{2}}\right\}\\ a=10\sqrt{2}\cos 42°\end{array}$

Write the expression to calculate the volume of the regular square pyramid.

  $V=\frac{1}{3}{a}^{2}h$  ... (1)

Where,

• $V$ is the volume of the pyramid.

Substitute the values in equation (1) and use calculator to solve the expression.

  $\begin{array}{c}V=\frac{1}{3}{\left(10\sqrt{2}\cos 42°\right)}^2\times 10\sin 42°\\ =\frac{1}{3}\times 110.45\times 6.69\\ =246.35\text{cubic}\text{unit}\\ \approx 246\text{cubic}\text{unit}\end{array}$

Conclusion:

Therefore, the volume of the regular square pyramid is $246\text{cubic}\text{unit}$ .