How To Find The Surface Area Of A Pyramid With A Rectangular Base

This article will guide you through the process using a straightforward algebraic formula, complete with a detailed example.

Formula to Calculate the Surface Area of a Pyramid with a Rectangular Base

The surface area ($SA$) of a pyramid with a rectangular base can be calculated using the following formula:

$$SA=L\cdot W+L\cdot \sqrt{{\left({\displaystyle \frac{W}{2}}\right)}^2+H^2}+W\cdot \sqrt{{\left({\displaystyle \frac{L}{2}}\right)}^2+H^2}$$

Where:

- $SA$ is the surface area of the pyramid.

- $L$ is the length of the rectangular base.

- $W$ is the width of the rectangular base.

- $H$ is the height of the pyramid (perpendicular height from the base to the apex).

Explanation of the Formula

The formula consists of three parts:

1. $L\cdot W$: This part calculates the area of the rectangular base.

2. $L\cdot \sqrt{{\left({\displaystyle \frac{W}{2}}\right)}^2+H^2}$: This part calculates the combined area of the two triangular faces that have the base's length ($L$) as one of their sides.

3. $W\cdot \sqrt{{\left({\displaystyle \frac{L}{2}}\right)}^2+H^2}$: This part calculates the combined area of the two triangular faces that have the base's width ($W$) as one of their sides.

Example Calculation

Let's go through an example to illustrate how to use this formula.

Given:

- $L=6$ units (the length of the rectangular base)

- $W=4$ units (the width of the rectangular base)

- $H=5$ units (the height of the pyramid)

We want to find the surface area of the pyramid.

Step-by-Step Calculation

Step 1: Identify the Given Values

Given:

- $L=6$ units

- $W=4$ units

- $H=5$ units

Step 2: Use the Surface Area Formula

$$SA=L\cdot W+L\cdot \sqrt{{\left({\displaystyle \frac{W}{2}}\right)}^2+H^2}+W\cdot \sqrt{{\left({\displaystyle \frac{L}{2}}\right)}^2+H^2}$$

Step 3: Substitute the Given Values into the Formula

$$SA=6\cdot 4+6\cdot \sqrt{{\left({\displaystyle \frac{4}{2}}\right)}^2+5^2}+4\cdot \sqrt{{\left({\displaystyle \frac{6}{2}}\right)}^2+5^2}$$

Step 4: Calculate the Area of the Rectangular Base

$$6\cdot 4=24$$

Step 5: Calculate the Area of the Triangular Faces

First triangular faces:

$$6\cdot \sqrt{{\left({\displaystyle \frac{4}{2}}\right)}^2+5^2}=6\cdot \sqrt{2^2+5^2}=6\cdot \sqrt{4+25}=6\cdot \sqrt{29}\approx 6\cdot 5.39=32.34$$

Second triangular faces:

$$4\cdot \sqrt{{\left({\displaystyle \frac{6}{2}}\right)}^2+5^2}=4\cdot \sqrt{3^2+5^2}=4\cdot \sqrt{9+25}=4\cdot \sqrt{34}\approx 4\cdot 5.83=23.32$$

Step 6: Sum the Three Parts to Find the Total Surface Area

$$SA=24+32.34+23.32=79.66$$

Final Value

The surface area of a pyramid with a rectangular base of length 6 units, width 4 units, and height 5 units is approximately $79.66$ square units.