Question

Find the surface area and volume of a rectangular pyramid.

Answer 1

Step 1: Definition of rectangular pyramid:

Pyramids are three-dimensional structures having triangle faces and have an encompassing polygon shape at their base.

A rectangular pyramid is simply a pyramid with a base that is a rectangle.

$V=$The volume of the rectangular pyramid

$A=$The total surface area of the rectangular pyramid

$L=$Length of the base

$B=$Width of the base

$H=$Height of the pyramid

$\mathcal{l}=$Slant height of the pyramid

Step 2: Solve for surface area of pyramid

The Lateral surface area of a rectangular pyramid is

$LSA=\frac{1}{2}\times$Perimeter of base $\times$ Slant height

We know that perimeter of rectangle $=2\left(L+B\right)$

Slant height $\mathcal{l}=\sqrt{H^2+{\left(\frac{L}{2}\right)}^2}$

$\therefore LSA=\frac{1}{2}\times 2\left(L+B\right)\times \sqrt{H^2+{\left(\frac{L}{2}\right)}^2}$

$\Rightarrow$$LSA=\left(L+B\right)\times \sqrt{H^2+{\left(\frac{L}{2}\right)}^2}$

Area of base $A\text{'}=L\times B$

$\therefore A=LSA+A\text{'}$

$\Rightarrow A=\left(L+B\right)\times \sqrt{H^2+{\left(\frac{L}{2}\right)}^2}+\left(L\times B\right)$

Step 3: Solve for volume of the pyramid

The volume of the rectangular pyramid $V=\frac{1}{3}\times$ Area of base $\times$ Height of the pyramid

$$\begin{gathered} \therefore V=\frac{1}{3}\times A\text{'}\times H \\ \Rightarrow V=\frac{L\times B\times H}{3} \end{gathered}$$

Hence, the surface area and volume of the rectangular pyramid are $\left(L+B\right)\times \sqrt{H^2+{\left(\frac{L}{2}\right)}^2}+\left(L\times B\right)$ and $\frac{L\times B\times H}{3}$ respectively