Surface Area Of A Square Pyramid

🔑 Key Concepts

• A square pyramid has one square base and four congruent triangular lateral faces.
• The base side length is used to find the area of the square base.
• The slant height is used as the height of each triangular lateral face.
• The formula \(SA = 2bs + b^2\) combines the area of the four triangular faces and the square base.
• In the formula, \(b\) represents the side length of the square base and \(s\) represents the slant height.
• The final answer should be written in square units because surface area measures total covering.

✏️ Worked Example

Square Pyramid with 6 cm Base and 12 cm Slant Height

Given:

Base side length: \(b = 6\text{ cm}\)

Slant height: \(s = 12\text{ cm}\)

Step 1: Use the square pyramid formula.

\(SA = 2bs + b^2\)

Step 2: Substitute the values.

\(SA = 2(6)(12) + 6^2\)

Step 3: Simplify each part.

\(2(6)(12) = 144\)

\(6^2 = 36\)

Step 4: Add the lateral area and base area.

\(SA = 144 + 36\)

\(SA = 180\)

Surface Area:

\(SA = 180\text{ cm}^2\)

🧠 Math Vocabulary

• surface area: the total area of all outside faces of a three-dimensional figure.
• square pyramid: a pyramid with a square base and four triangular lateral faces.
• base: the square face of the pyramid.
• lateral face: one of the triangular faces that meets at the top vertex of the pyramid.
• slant height: the height of one triangular lateral face, measured along the face of the pyramid.
• base side length: the side length of the square base, represented by \(b\) in the formula.
• square units: units used to measure area, such as \( \text{cm}^2 \), \( \text{in}^2 \), or \( \text{ft}^2 \).

💡 Main Idea

The surface area of a square pyramid can be found by adding the area of the square base and the area of the four triangular lateral faces. When the base side length and slant height are known, the formula \(SA = 2bs + b^2\) is a compact way to organize the same calculation.

📚 What You Should Already Know

Students should already know how to find the area of a square, calculate the area of a triangle, substitute values into a formula, and simplify expressions involving multiplication and exponents.

🚀 What Comes Next

After using the square pyramid formula, students can compare formula-based strategies with net-based strategies. They can also extend surface area work to other solids, composite figures, and real-world problems involving packaging, covering, and construction.

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