Surface Area - Cubes, Rectangular Prisms, Triangular Prisms And Square Pyramids

🔑 Key Concepts

• Surface area is the total area of all outside faces of a three-dimensional figure.
• Nets allow us to visualize all faces of a solid at one time.
• Cube surface area can be found using \(6s^2\).
• Rectangular prisms contain three pairs of congruent rectangles.
• Triangular prisms contain three rectangles and two congruent triangles.
• Square pyramids contain one square base and four triangular faces.
• Always include square units in your final answer.

✏️ Worked Examples

Example from the Video: Cube Net

Step 1: Find the area of one square face.

\( 5.5 \times 5.5=30.25 \)

Step 2: Multiply by the six faces of the cube.

\( 30.25\times6=181.5 \)

Surface Area:

\( 181.5\text{ in}^2 \)

Example from the Video: Square Pyramid Net

Area of one triangle:

\( \frac{6\times6}{2}=18 \)

Four triangular faces:

\( 18\times4=72 \)

Square base:

\( 6\times6=36 \)

Total surface area:

\( 72+36=108 \)

\( 108\text{ units}^2 \)

Example from the Video: Rectangular Prism

Find the three unique face areas:

\( 5\times8=40 \)

\( 5\times9=45 \)

\( 8\times9=72 \)

Add them:

\( 40+45+72=157 \)

Double because each face appears twice:

\( 157\times2=314 \)

\( 314\text{ cm}^2 \)

Example from the Video: Triangular Prism Net

Find the rectangular faces:

\( 5\times8=40 \)

\( 6\times8=48 \)

\( 5\times8=40 \)

Total rectangular area:

\( 40+48+40=128 \)

Find the two triangles:

\( \frac{6\times4}{2}=12 \)

\( 12\times2=24 \)

Total surface area:

\( 128+24=152 \)

\( 152\text{ ft}^2 \)

🧠 Math Vocabulary

• surface area: total area of all outside faces.
• net: a two-dimensional representation of a three-dimensional figure.
• face: a flat surface of a solid.
• edge: a segment where two faces meet.
• vertex: a point where edges meet.
• lateral face: any face that is not a base.
• base: the congruent faces of a prism or the polygon foundation of a pyramid.
• cube: a prism with six congruent square faces.
• rectangular prism: a prism with rectangular faces.
• triangular prism: a prism with two triangular bases.
• square pyramid: a pyramid with a square base.
• slant height: the height of a triangular face of a pyramid.

💡 Main Idea

Surface area problems require identifying every exposed face of a three-dimensional figure, calculating the area of each face, and adding those areas together. Nets help visualize the surfaces and are especially useful when solving problems involving prisms, pyramids, and cubes.

📚 What You Should Already Know

Students should know how to calculate the area of rectangles and triangles, multiply decimals and whole numbers, identify two-dimensional shapes, and understand basic properties of prisms and pyramids.

🚀 What Comes Next

After mastering surface area using nets and prisms, students can explore composite solids, cylinders, cones, and the relationship between surface area and volume.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning management system.