Surface Area Of Various Nets

🔑 Key Concepts & Formulas

• Surface area is found by adding the areas of all outside faces of a three-dimensional figure.
• A net shows the faces of a solid laid flat so each individual surface can be counted and measured.
• For a cube, all 6 faces are congruent squares.
• For a triangular prism, add the areas of 3 rectangular faces and 2 congruent triangular bases.
• For a square pyramid, add the area of the square base and the 4 triangular lateral faces.
• The final answer must use square units because surface area measures two-dimensional covering.

📐 Formal Surface Area Formulas

In this lesson, students focus on finding the area of each individual surface and adding the pieces together. As students become more fluent, this same thinking can be formalized using surface area formulas.

• Cube: \(SA = 6s^2\)
• Rectangular Prism: \(SA = 2lw + 2lh + 2wh\)
• Triangular Prism: \(SA = 2B + PH\), where \(B\) is the area of one triangular base, \(P\) is the perimeter of the triangular base, and \(H\) is the height (length) of the prism.
• Square Pyramid: \(SA = b^2 + 2bs\), where \(b\) is the side length of the square base and \(s\) is the slant height of the triangular faces.

✏️ Worked Examples

Jewelry Box Cube Net

Problem: A cube-shaped jewelry box has side length \(3.5\) inches. How much paper is needed to cover it?

Area of one square face:

\( 3.5 \times 3.5 = 12.25 \)

A cube has 6 congruent square faces:

\( 12.25 \times 6 = 73.5 \)

Surface Area:

\( 73.5\text{ in}^2 \)

Triangular Prism Net

Problem: Find the surface area of the triangular prism net.

Two triangular bases:

\( \frac{6 \times 4}{2} = 12 \)

\( 12 \times 2 = 24 \)

Three rectangular faces:

\( 5 \times 9 = 45 \)

\( 6 \times 9 = 54 \)

\( 5 \times 9 = 45 \)

Total surface area:

\( 24 + 45 + 54 + 45 = 168 \)

\( 168\text{ ft}^2 \)

Square Pyramid Net

Problem: Find the surface area of the square pyramid net.

Area of one triangular face:

\( \frac{8 \times 6}{2} = 24 \)

Four triangular faces:

\( 24 \times 4 = 96 \)

Square base:

\( 8 \times 8 = 64 \)

Total surface area:

\( 96 + 64 = 160 \)

\( 160\text{ ft}^2 \)

🧠 Math Vocabulary

• surface area: the total area of all outside faces of a three-dimensional figure.
• net: a flat pattern that can be folded into a three-dimensional figure.
• face: a flat surface of a solid figure.
• lateral face: a face that is not a base.
• base: a reference face of a solid; prisms have two congruent bases, while pyramids have one base.
• cube: a prism with 6 congruent square faces.
• triangular prism: a prism with 2 triangular bases and 3 rectangular lateral faces.
• square pyramid: a pyramid with a square base and 4 triangular lateral faces.
• slant height: the height of a triangular lateral face of a pyramid.
• square units: units used to measure area, such as \( \text{in}^2 \), \( \text{ft}^2 \), \( \text{cm}^2 \), or \( \text{units}^2 \).

💡 Main Idea

Surface area can be found by decomposing a three-dimensional figure into familiar two-dimensional shapes. Instead of memorizing first, students can use nets to see each individual surface, calculate each area, and add those areas together to find the total covering.

📚 What You Should Already Know

Students should already know how to calculate the area of squares, rectangles, and triangles. They should also be comfortable multiplying whole numbers and decimals, using the triangle area formula \(A=\frac{1}{2}bh\), and labeling area answers with square units.

🚀 What Comes Next

After students understand surface area as the sum of individual faces, they can connect that visual strategy to formal formulas for cubes, rectangular prisms, triangular prisms, and pyramids. This helps students move from counting and adding faces in a net to using efficient formulas for more complex surface area problems.

🧩 Embed This Video in Your LMS

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