Represent 3D Figures Using Nets

Nets and Surface Area

In this lesson:

• Identify nets of 3D figures
• Find surface area using nets
• Apply surface area to real-world problems

Three Goals for This Lesson

By the end, you should be able to:

1. Identify the net of a given 3D figure by visualizing or folding
2. Find surface area of a prism by summing face areas
3. Apply surface area to real-world problems

From 3D Box to Flat Net

You already know:

• A rectangular prism has 6 faces — 3 pairs of congruent rectangles
• Area of a rectangle = length × width

Imagine cutting the edges of a cardboard box and unfolding it flat — that flat shape is the net.

What Is a Net? Unfolding 3D Shapes

A net is a flat, 2D figure that folds into a 3D shape — every face appears exactly once.

The Net of a Rectangular Prism

• 6 faces total: 3 congruent pairs
• Top/Bottom: each is
• Front/Back: each is
• Left/Right: each is

Surface Area from the Net

Example: Prism with , ,

Compute each face area, then sum:

Surface area = 48 + 36 + 24 = 108 sq units

The Surface Area Formula for Prisms

Verify (, , ):

• SA uses square units (area of faces)
• Volume uses cubic units — a different measure

Quick Check: All 6 Faces

A prism has , , .

A student computes:
, , , total = 47 sq units.

What did the student forget?

Think before the next slide…

Answer: Double Each Face Pair

The student forgot to double each face pair — the net has 6 faces, not 3.

Correct SA = 30 + 40 + 24 = 94 square units

Checking Valid vs. Invalid Nets

• Valid net: all 6 faces present, no overlap when folded
• Invalid net: faces overlap, or fewer than 6 panels

Check: Count first — a rectangular prism always needs exactly 6 rectangles. Then mentally fold to verify no overlaps occur.

The Triangular Prism Has Five Faces

• 5 faces: 2 triangular bases + 3 rectangular sides
• Each rectangle width = one side of the triangle

Computing Surface Area of Triangular Prisms

Right-triangle base: legs 3, 4; hypotenuse 5; prism length = 8

Triangular faces:

Rectangular faces:

Watch Out: Three Separate Measurements

Misconception: using the prism length as the triangle's height.

The triangular prism has three separate dimensions:

The prism length does NOT go into the triangle area formula.

Gift Box: Full Surface Area Needed

Gift box: 10 in × 6 in × 3 in. Wrapping paper covers all 6 faces.

Tent Tarp: Partial Surface Area

Tent: legs 3, 4; hyp 5; length 7. Cover 2 slanted sides + 1 triangle end.

Key Takeaways for Nets and Surface Area

• Net: flat 2D figure that folds into a 3D shape
• Surface area = sum of all face areas

Watch for:

• Missing faces — double each pair (×2)
• SA in square units; volume in cubic units
• Prism length ≠ triangle height

What You Can Do Next

You can now:

• Identify nets and verify they fold correctly
• Compute surface area for prisms
• Solve full and partial surface area problems

Up next: Practice problems — surface area of prisms in real-world contexts.