Nets and Surface Area of Prisms

In this lesson:

• Unfold a 3D figure into a flat net
• Find the surface area of a rectangular prism
• Tell surface area apart from volume

What You Will Be Able to Do

By the end, you will:

1. Count faces, edges, and vertices of a prism
2. Unfold a 3D figure into a 2D net
3. Decide whether an arrangement is a valid net
4. Compute surface area by summing face areas

How Much Wrapping Paper Covers This Box?

A gift box is 4 in long, 3 in wide, 2 in tall.

You need paper to cover top, bottom, front, back, and both sides.

• A drawing only shows three faces — what about the others?
• How do you count every face?

Unfold the Box: Every Face in Plain View

Imagine the box's flaps swinging open like hinges:

1. The lid lifts up
2. The four side walls fall outward
3. The bottom stays in place

What's left is 6 flat rectangles arranged in a cross.

Now you can see — and measure — every face at once.

A Net Folds Back into the Figure

A net is a 2D pattern of polygons that folds back into the 3D figure.

Four Figures Made of Rectangles and Triangles

Each figure has a fingerprint: count the faces, edges, and vertices.

One Figure, Many Valid Nets

A cube has 11 different valid nets — all fold into the same cube.

You can hinge faces apart in many orders:

• A "T" arrangement
• A "+" arrangement
• A staircase

Any edge-to-edge unfolding is valid.

Predict: Is This a Cube Net?

A flat arrangement has 6 squares, all the same size, in this layout:

    [ ]
[ ][ ][ ][ ]
[ ]

Does it fold into a cube? Yes or No?

Commit to your answer before the next slide.

Net Validation: Two Things to Check

For a candidate net to be valid:

1. Right face count and shapes — match the target figure
2. Folds without overlap or gaps — every face lands in its place

The arrangement on the previous slide is valid — it's one of the 11 cube nets.

Match Each Figure to Its Net

Match each figure by faces and shapes:

Count, then break ties by shape.

From "Name the Figure" to "Measure the Figure"

You can now:

• Recognize each figure
• Unfold it into a net
• Validate a candidate net

Next question: how much area covers a prism's outside?

The net does the heavy lifting — each face is a rectangle.

Surface Area Is the Sum of Face Areas

Definition: Surface area = sum of all face areas.

Measured in square units — square in, square cm, square ft.

Worked Example: Surface Area of a 4×3×2 Prism

• Top + bottom:
• Front + back:
• Left + right:

Regroup the Sum into a Formula

Six faces come in 3 pairs:

• Top & bottom: each — total
• Front & back: each — total
• Left & right: each — total

Worked Example: 5×4×3 Prism with Formula

Identify , , . Substitute:

Cube Shortcut for Equal Side Lengths

A cube is a rectangular prism with all sides equal.

For a 6×6×6 cube: all 6 faces are squares.

General rule: for any cube of side .

Fractional Edges: Same Rule, Careful Arithmetic

A prism with , , :

Surface Area vs Volume: Same Cube, Two Answers

Guided Practice: Fill in the Pair Products

A 3×4×5 prism. Fill in each pair:

• Top/bottom: ; pair $= $ ___
• Front/back: ; pair $= $ ___
• Left/right: ; pair $= $ ___

Check-In: Surface Area of a 5×5×5 Cube

Compute the surface area. Write your answer before advancing.

• One face area: ___
• Six faces total: ___ sq units

Commit to one number before advancing.

Your Turn: Sketch the Net Yourself

A rectangular prism: , , .

1. Sketch the net (any valid arrangement)
2. Label each rectangle's dimensions
3. Compute each face area, then sum

No formula. Build it from the net.

Answer and Common Errors to Avoid

square units.

Wrong: (volume)
Wrong: (forgot to double)
✓ Right:

Key Takeaways and Common Warnings

✓ Net = 2D unfolding that folds back

✓ SA = sum of face areas, in square units

✓ Rectangular prism:

Volume = cubic units; SA = square units

Six faces in three pairs — don't forget doubling

Next Lesson: Triangular Prisms, Pyramids, and Real-World Applications

You can now unfold any rectangular prism and find its surface area.

In Lesson 2:

• Triangular prisms — 2 triangles + 3 rectangles
• Square pyramids — using slant height
• Real-world problems — wrap, tents, paint