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:
\[ SA = L \cdot W + L \cdot \sqrt{\left(\dfrac{W}{2}\right)^2 + H^2} + W \cdot \sqrt{\left(\dfrac{L}{2}\right)^2 + H^2} \]
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. \( L \cdot \sqrt{\left(\dfrac{W}{2}\right)^2 + H^2} \): This part calculates the combined area of the two triangular faces that have the base's length (\(L\)) as one of their sides.
3. \( W \cdot \sqrt{\left(\dfrac{L}{2}\right)^2 + H^2} \): 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
\[ SA = L \cdot W + L \cdot \sqrt{\left(\dfrac{W}{2}\right)^2 + H^2} + W \cdot \sqrt{\left(\dfrac{L}{2}\right)^2 + H^2} \]
Step 3: Substitute the Given Values into the Formula
\[ SA = 6 \cdot 4 + 6 \cdot \sqrt{\left(\dfrac{4}{2}\right)^2 + 5^2} + 4 \cdot \sqrt{\left(\dfrac{6}{2}\right)^2 + 5^2} \]
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:
\[ 6 \cdot \sqrt{\left(\dfrac{4}{2}\right)^2 + 5^2} = 6 \cdot \sqrt{2^2 + 5^2} = 6 \cdot \sqrt{4 + 25} = 6 \cdot \sqrt{29} \approx 6 \cdot 5.39 = 32.34 \]
Second triangular faces:
\[ 4 \cdot \sqrt{\left(\dfrac{6}{2}\right)^2 + 5^2} = 4 \cdot \sqrt{3^2 + 5^2} = 4 \cdot \sqrt{9 + 25} = 4 \cdot \sqrt{34} \approx 4 \cdot 5.83 = 23.32 \]
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.
