Lesson 6.6 — Build Open Boxes and Compare Volumes
立体图形问题：无盖纸盒与容积比较
Turn a flat sheet into an open box, track how every cut changes the height and base, and compare designs without trusting appearances.
See how a flat square becomes an open box
Treat the sheet as having negligible thickness, with no extra material for seams. All lengths are in cm and volumes in cm³ unless stated otherwise. Start with a square sheet. Cut the same small square from all four corners, then fold the four side flaps upward.
Begin with a 12 cm square sheet.
What becomes the height?
The side length of each removed corner square becomes the height of the box.
height = xWhat becomes the base?
Each base dimension loses one cut from the left and one from the right.
base side = s − 2xWhy is there no lid?
The four flaps make side walls only. No face is folded over the top.
Track the dimensions
A 12 cm square sheet has 2 cm squares removed from its corners.
Build the open-box formula from the dimensions
Let the original square sheet have side length s, and let every corner cut have side length x.
Use the formula family
For the last two answers use s = 14 cm and x = 3 cm, regardless of the live model.
Compare four designs using 12 cm square sheets
Use a separate identical 12 cm square sheet for each design. Only the corner-cut size changes.
| Design | Cut | Base | Height | Volume |
|---|---|---|---|---|
| A | 4 | 4×4 | 4 | 64 |
| B | 3 | 6×6 | 3 | 108 |
| C | 2 | 8×8 | 2 | 128 |
| D | 1 | 10×10 | 1 | 100 |
Largest volume: C, 128 cm³.
Complete the comparison
Understand the trade-off: taller box, smaller base
A larger corner cut increases the height, but it shrinks both dimensions of the square base.
Cut gets larger
x ↑
Height increasesBase side gets smaller
12−2x ↓
Both base dimensions decreaseExplain the changing volume
Check whether a cut makes a real box
The cut side length x must be positive and less than half the sheet side length s. Otherwise the box has zero height or no positive base side.
Forward reasoning
Given sheet side s and cut x:
base side = s−2xBackward reasoning
Given base side b and height h:
sheet side = b+2hWork in both directions
Use a design table instead of guessing
For a whole-centimetre cut size, list every positive whole number x less than half the sheet side, calculate each volume, and compare.
| Cut x | Base side s−2x | Base area | Volume | Comparison |
|---|
18 cm sheet challenge
Use an 18 cm sheet for these answers, regardless of the studio slider.
Fold an L-shaped sheet into a square-based open box
Use the specific L-shaped net shown: a strip of four identical rectangular side panels in a row, with one square base flap attached above an end panel. The strip has total length H and height h; each panel and the square flap have width q. Thus H = 4q and the sheet’s total vertical length is V = q + h. In Missions 7–10, V labels this vertical length, not volume.
q=H÷4
h=V−q
q²h
Use sheet A: H = 120 cm and V = 80 cm
Compare three L-shaped sheets
Longer horizontal material makes a larger square base, but it may leave less height.
| Sheet | H | V | Base side H÷4 | Height V−H÷4 | Volume |
|---|---|---|---|---|---|
| A | 120 | 80 | 30 | 50 | 45,000 |
| B | 140 | 75 | 35 | 40 | 49,000 |
| C | 160 | 70 | 40 | 30 | 48,000 |
What grows?
From A to C, the square base side grows from 30 to 40 cm.
What shrinks?
The height falls from 50 to 30 cm. Sheet B gives the best balance.
Complete the comparison
Open-box design workshop
Calculate carefully. You need at least 6 of 8 correct to complete the workshop.
A 14 cm square has 2 cm corner cuts. Find the volume.
Hint
Base side: 14−4=10.
A 16 cm square has 3 cm corner cuts. Find the volume.
Hint
Use 3(16−6)^2.
A 20 cm square has 4 cm corner cuts. Find the volume.
Hint
The base is 12×12.
For an 18 cm square, what is the largest volume among integer cuts?
Hint
Compare cuts 1 through 8.
Sheet A: H=120, V=80. Find the volume.
Hint
q=30, h=50.
Sheet B: H=140, V=75. Find the volume.
Hint
q=35, h=40.
Sheet C: H=160, V=70. Find the volume.
Hint
q=40, h=30.
An L-sheet with Mission 7’s layout has H=100 cm and V=45 cm. Find its box volume.
Hint
q=25, h=20.
Exit ticket
Answer all five objective questions. A perfect result unlocks your certificate.
1 · Base side
A 12 cm square has 3 cm corner cuts. What is the base side?
2 · Volume
What is the volume of that box?
3 · Compare designs
Among the four 12 cm sheet designs in Mission 3, which has the largest volume?
4 · L-shaped sheet
For H=140 and V=75, what is the box height?
5 · Design table
For an 18 cm sheet, which integer cut gives the largest volume?
Optional reflection
Not automatically graded. Explain why the tallest box need not have the largest volume.
Certificate of mastery
Open-Box Design Engineer
This certifies that a determined mathematician can turn flat sheets into open boxes, calculate volume, and compare competing designs.
Lesson 6.6 · Grade 5 Math Explorer
Teaching notes
The four 12 cm square-sheet designs and their volumes are based on Chapter 6, Example 6. The three L-shaped sheets with dimensions 120×80, 140×75, and 160×70 are based on the guided practice immediately following it.
Lesson summary
Complete the missions to become a careful open-box designer.