Square Pyramid Surface Area

🔑 Key Concepts

• A square pyramid has one square base and four triangular faces.
• A net shows all five faces of the square pyramid laid flat.
• To find surface area, calculate the area of the square base and the area of each triangular face.
• If the four triangular faces are congruent, you can find the area of one triangle and multiply by 4.
• The height shown inside each triangular face is the slant height, not the vertical height inside the pyramid.
• The final answer must be written in square units because surface area measures covering.

✏️ Worked Example

Square Pyramid Net with 7 Unit Base and 6 Unit Slant Height

Identify the faces:

1 square base

4 congruent triangular faces

Step 1: Find the area of the square base.

\(7 \times 7 = 49\)

Step 2: Find the area of one triangle.

\(\frac{7 \times 6}{2}\)

\(\frac{42}{2}=21\)

Each triangular face has an area of \(21\text{ square units}\).

Step 3: Find the area of all four triangles.

\(21 \times 4 = 84\)

Step 4: Add the base area and triangular area.

\(49 + 84 = 133\)

Surface Area:

\(SA = 133\text{ square units}\)

🧠 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.
• net: a flat pattern that can be folded to form a three-dimensional figure.
• base: the bottom face of the pyramid; in this lesson, the base is a square.
• lateral face: a face that is not the base; for a square pyramid, the lateral faces are triangles.
• slant height: the height of a triangular lateral face, measured from the base of the triangle to the top vertex.
• congruent: having the same size and shape.
• square units: units used to measure area, such as \( \text{cm}^2 \), \( \text{in}^2 \), \( \text{ft}^2 \), or \( \text{units}^2 \).

💡 Main Idea

To find the surface area of a square pyramid net, break the net into one square and four triangles. Find the area of the square base, find the area of the triangular faces, and add everything together to get the total surface area.

📚 What You Should Already Know

Students should know how to find the area of a square, how to find the area of a triangle using \(A=\frac{1}{2}bh\), and how to add multiple areas together. It also helps to recognize that a net is made from two-dimensional shapes.

🚀 What Comes Next

After students can calculate surface area from a square pyramid net, they can compare this strategy to other solids such as cubes, rectangular prisms, triangular prisms, cylinders, and composite figures. This builds toward choosing efficient strategies for different three-dimensional figures.

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