Surface Area Of A Rectangular Prism Using A Net

🔑 Key Concepts

• A rectangular prism has 6 rectangular faces.
• Opposite faces are congruent, so each unique face area appears twice.
• A net helps show all the faces of a three-dimensional figure as a two-dimensional layout.
• The formula \(SA=2lw+2lh+2wh\) matches the three pairs of congruent faces.
• Surface area is measured in square units.

✏️ Worked Examples

In this example, the rectangular prism has dimensions \(L=9\), \(W=6\), and \(H=4\). The net highlights one pair of congruent faces at a time.

1. Right and Left Faces

Right and left faces:

\(L \times H\)

\(9 \times 4 = 36\)

There are 2 congruent faces:

\(2(9 \times 4)=2(36)=72\)

2. Top and Bottom Faces

Top and bottom faces:

\(L \times W\)

\(9 \times 6 = 54\)

There are 2 congruent faces:

\(2(9 \times 6)=2(54)=108\)

3. Front and Back Faces

Front and back faces:

\(W \times H\)

\(6 \times 4 = 24\)

There are 2 congruent faces:

\(2(6 \times 4)=2(24)=48\)

Put the three pairs together:

\(SA=2lh+2lw+2wh\)

\(SA=2(9\times4)+2(9\times6)+2(6\times4)\)

\(SA=72+108+48\)

\(SA=228\text{ square units}\)

🧠 Math Vocabulary

• Rectangular Prism: a three-dimensional figure with six rectangular faces.
• Net: a two-dimensional pattern that can be folded to make a three-dimensional figure.
• Face: one flat surface of a three-dimensional figure.
• Surface: the outside part or face of a three-dimensional figure.
• Surface Area: the total area of all faces on the outside of a three-dimensional figure.
• Congruent Faces: faces that have the same size and shape.
• Edge: a line segment where two faces meet.
• Vertex: a point where edges meet.
• Plane: a flat two-dimensional surface that extends in all directions.
• Length, Width, and Height: the three dimensions used to describe a rectangular prism.
• Square Units: units used to measure area, such as \(\text{cm}^2\), \(\text{ft}^2\), or \(\text{units}^2\).

💡 Main Idea

A rectangular prism has three pairs of congruent faces. To find surface area, find the area of each unique face pair, double each area, and add the results.

📚 What You Should Already Know

Students should already know how to find the area of rectangles, multiply whole numbers, identify length, width, and height, and recognize that a net represents the faces of a three-dimensional figure.

🚀 What Comes Next

Students can extend this idea to surface area problems involving cubes, triangular prisms, cylinders, and composite three-dimensional figures.

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