Surface Area - A Pyramid On Top Of A Cube

🔑 Key Concepts

• A composite solid is made by combining two or more three-dimensional figures.
• Only count the outside faces that are visible or exposed.
• The shared square between the cube and pyramid is not part of the outside surface area.
• The cube contributes 5 square faces because the top face is covered by the pyramid.
• The square pyramid contributes 4 congruent triangular faces.

✏️ Worked Examples

Step 1: Cube Surface Area

The cube has side length \(3\text{ cm}\).
One square face has area:
\(3\times3=9\text{ cm}^2\)

The top face is covered by the pyramid, so count only 5 square faces:
\(5(9)=45\text{ cm}^2\)

Step 2: Pyramid Lateral Faces

The pyramid has 4 congruent triangular faces.

Each triangle has base \(3\text{ cm}\) and slant height \(2.5\text{ cm}\).

Area of one triangle:
\(\displaystyle \frac{3\times2.5}{2}=3.75\text{ cm}^2\)

Area of 4 triangles:
\(4(3.75)=15\text{ cm}^2\)

Step 3: Add Exposed Surfaces

Add the exposed surface area from the cube and the pyramid.

\(45+15=60\)

The surface area of the composite figure is:
\(60\text{ cm}^2\)

🧠 Math Vocabulary

• Surface area: The total area of all exposed outside faces of a three-dimensional figure.
• Composite solid: A three-dimensional figure made by combining two or more solids.
• Cube: A prism with 6 congruent square faces. Surface area formula: \(SA=6s^2\), where \(s\) is the side length.
• Square pyramid: A pyramid with a square base and 4 triangular lateral faces. Surface area formula: \(SA=s^2+2s\ell\), where \(s\) is the base side length and \(\ell\) is the slant height.
• Lateral area: The area of the side faces of a solid, not including the base or bases.
• Congruent surfaces: Faces that have the same size and same shape.
• Apex: The top vertex of a pyramid where the triangular faces meet.
• Vertex: A corner point where edges meet. The plural form is vertices.
• Edge: A line segment where two faces meet.

💡 Main Idea

To find the surface area of a composite solid, break the figure into familiar shapes, calculate the area of each exposed surface, and avoid counting any hidden or shared faces.

📚 What You Should Already Know

Students should know how to find the area of squares and triangles, identify faces of three-dimensional figures, and use surface area formulas for prisms and pyramids.

🚀 What Comes Next

After learning surface area of composite solids, students can extend this thinking to more complex figures, nets, volume, and real-world design problems involving packaging or construction.

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