In this article, we will guide you through the process of calculating the surface area of a pyramid with a square base. The surface area includes the area of the square base and the area of the four triangular faces.
Step-by-Step Guide
Step 1: Show the Surface Area Formula
The surface area (SA) of a pyramid with a square base can be found using the following formula:
\[ SA = a^2 + a \cdot \sqrt{a^2 + 4 \cdot h^2} \]
Where:
- \( a \) is the length of a side of the square base.
- \( h \) is the height of the pyramid from the center of the base to the apex.
Step 2: Explain the Formula
- The term \( a^2 \) represents the area of the square base.
- The term \( a \cdot \sqrt{a^2 + 4 \cdot h^2} \) represents the combined area of the four triangular faces of the pyramid.
Step 3: Insert Numbers as an Example
Let's consider a pyramid with a square base where:
- Side length of the square base: \( a = 4 \) units
- Height of the pyramid: \( h = 6 \) units
Step 4: Calculate the Final Value
First, calculate the area of the square base:
\[ \text{Area of the square base} = a^2 = 4^2 = 16 \, \text{square units} \]
Next, calculate the area of the four triangular faces:
\[ \text{Area of the triangular faces} = a \cdot \sqrt{a^2 + 4 \cdot h^2} \]
\[ = 4 \cdot \sqrt{4^2 + 4 \cdot 6^2} \]
\[ = 4 \cdot \sqrt{16 + 4 \cdot 36} \]
\[ = 4 \cdot \sqrt{16 + 144} \]
\[ = 4 \cdot \sqrt{160} \]
\[ = 4 \cdot \sqrt{16 \cdot 10} \]
\[ = 4 \cdot 4 \cdot \sqrt{10} \]
\[ = 16 \cdot \sqrt{10} \approx 16 \cdot 3.162 \approx 50.59 \, \text{square units} \]
Finally, add the areas of the square base and the triangular faces to find the total surface area:
\[ SA = 16 + 50.59 \approx 66.59 \, \text{square units} \]
Final Value
The surface area of a pyramid with a square base, where the side length of the base is 4 units and the height of the pyramid is 6 units, is approximately 66.59 square units.
