A storm is moving toward Prism City. Kira, Theo, Nova, and Engineer Sol must cover every face of the city signal before the first lightning strike. Their only map is a net, and every square unit matters.
KiraBuilder. Spots the faces that need panels.
TheoMap reader. Turns boxes into nets.
NovaArtist. Labels every rectangle clearly.
Engineer SolSignal keeper. Checks the final plan.
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Chapter 1: Alarm at Prism City
Set the mission
At 3:15, the city signal flickered. The storm glass turned red.
Engineer Sol: The signal needs a protective cover on every outside face.
Kira: Every face? Front, back, sides, top, and bottom?
FLASH
A supply crate rolled into the plaza.
Theo: The cover panels are packed in a rectangular prism.
Nova: Then our first clue is the size of the box: 6 by 4 by 3.
The team opened the case file.
Mission: Find the surface area so the crew knows how many square units of covering are needed.
means the total area of all outside faces of a 3D figure.
Chapter 2: The Net on the Table
See the faces
6 x 4
6 x 4
6 x 3
4 x 3
4 x 3
6 x 3
Theo unfolded the prism into a flat net.
Theo: A has 6 , but they come in 3 matching pairs.
Nova: I see two 6 by 4 faces, two 6 by 3 faces, and two 4 by 3 faces.
Kira tapped the matching rectangles.
Kira: Opposite faces match. We can find one area, then double it.
: top/bottom, front/back, left/right.
Your field check
How many matching pairs of faces does a rectangular prism have?
Chapter 3: The First Pair
Multiply side lengths
The top panel was marked 6 by 4.
Nova: Area of a rectangle is length times width.
Kira: So one 6 by 4 face is 24 square units.
Panel Ledger
6 x 4
24
6 x 4
24
Total so far
48
Two matching faces meant two panels of the same size.
Theo: Double the area because the opposite face matches.
Your turn in the ledger
What is the area of one 6 by 4 face?
Chapter 4: Six Faces, One Formula
Organize the total
Crate: 6 by 4 by 3
2(6 x 4)
48
2(6 x 3)
36
2(4 x 3)
24
Total
108
Engineer Sol checked each pair before the panels left the shop.
Engineer Sol: The short way is SA = 2lw + 2lh + 2wh.
Theo: But the formula only works if we know what each letter means.
No mystery letters
l = 6, w = 4, h = 3.
2(6 x 4) + 2(6 x 3) + 2(4 x 3) = 108 square units.
Your signal check
What is the total surface area of the 6 by 4 by 3 crate?
Chapter 5: The Final Crate
Apply the method
A second crate arrived during the storm.
Kira: This one is 8 by 5 by 2. If we rush, we might miss a face.
Nova: Pair it first. Then add.
Second Crate
2(8 x 5)
80
2(8 x 2)
32
2(5 x 2)
20
Total
132
The ledger filled with clean evidence.
Theo: The labels saved us from mixing up perimeter and area.
Write like a mathematician
Explain why the second crate needs 132 square units of covering.
Chapter 6: Signal Restored
Close the case
The final panels snapped into place as the storm split around Prism City.
Engineer Sol: The signal is covered. Your surface area plan saved the system.
Kira: We did not just use a formula. We proved where every number came from.
Case notes
Find surface area by drawing or imagining the net, finding each rectangular face area, pairing opposite faces, then adding all six faces.
Exit ticket
How is surface area different from volume?
Net
A flat pattern of all faces of a 3D figure.
Face Area
For each rectangular face, multiply the two side lengths.
Surface Area
Add the areas of all outside faces. Units are square units.