Calculating the surface area of a cuboid is essential for many real-world applications. This article will explain how to find the surface area using a simple algebraic formula, with detailed steps and an example.
Formula to Calculate the Surface Area of a Cuboid
The surface area \( SA \) of a cuboid can be calculated using the following formula:
\[ SA = 2 \cdot L \cdot W + 2 \cdot L \cdot D + 2 \cdot W \cdot D \]
Where:
- \( SA \) is the surface area of the cuboid.
- \( L \) is the length of the cuboid.
- \( W \) is the width of the cuboid.
- \( D \) is the depth (or height) of the cuboid.
Explanation of the Formula
A cuboid has six rectangular faces:
- Two faces are \( L \times W \).
- Two faces are \( L \times D \).
- Two faces are \( W \times D \).
The total surface area is the sum of the areas of all six faces, which is why the formula includes each pair of faces being calculated twice.
Example Calculation
Let's go through an example to illustrate how to use this formula.
Given:
- \( L = 5 \) units (the length of the cuboid)
- \( W = 3 \) units (the width of the cuboid)
- \( D = 4 \) units (the depth of the cuboid)
We want to find the surface area of the cuboid.
Step 1: Identify the Given Values
Given:
- \( L = 5 \) units
- \( W = 3 \) units
- \( D = 4 \) units
Step 2: Use the Surface Area Formula
\[ SA = 2 \cdot L \cdot W + 2 \cdot L \cdot D + 2 \cdot W \cdot D \]
Step 3: Substitute the Given Values into the Formula
\[ SA = 2 \cdot 5 \cdot 3 + 2 \cdot 5 \cdot 4 + 2 \cdot 3 \cdot 4 \]
Step 4: Calculate the Individual Products
\[ 2 \cdot 5 \cdot 3 = 30 \]
\[ 2 \cdot 5 \cdot 4 = 40 \]
\[ 2 \cdot 3 \cdot 4 = 24 \]
Step 5: Sum the Products to Find the Total Surface Area
\[ SA = 30 + 40 + 24 \]
\[ SA = 94 \]
Final Value
The surface area of a cuboid with dimensions \( 5 \) units by \( 3 \) units by \( 4 \) units is \( 94 \) square units.
By following these steps, you can easily calculate the surface area of a cuboid when you know its length, width, and depth. This method involves calculating the area of each pair of faces and summing them to get the total surface area.
