Problem
Ribbon parallel to an edge is as long as that edge
A box is tied with ribbon that goes around it three times one way and once the other. We need the total length of ribbon, ignoring the knot.
GeometryMeasurement & data
Your answer
How to solve
Strategy Draw a Diagram — Follow one band all the way round and see which edges it copies. Then only the number of bands has to be counted, and the hidden runs underneath the box are as easy to account for as the visible ones.
1STEP 1
Follow one band of the three
A band running up the front goes over the top, down the back and under the bottom, so it copies two heights and two depths.
80 × 2 + 90 × 2 = 340
Each of those bands is 340 cm.
6.G.A.4Draw A DiagramEach of the three bands is long.
Why?
A band lies flat on four faces and comes back to where it started, so its length is those four straight pieces added.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Opposite faces of a box are congruent, so the band crosses the same twice and the same twice.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
2STEP 2
Count the three bands
Three bands run that way, all the same length.
340 × 3 = 1020
1020 cm so far.
4.MD.A.3Identify Subproblems3STEP 3
Follow the crossing band
The band going the other way crosses the top along the length, drops down one side, runs back underneath and climbs the far side, copying two lengths and two heights.
110 × 2 + 80 × 2 = 380
That band is 380 cm.
6.G.A.4Visualize Spatial Relationships4STEP 4
Add everything
Add the three bands and the crossing band.
1020 + 380 = 1400
1400 cm of ribbon.
4.MD.A.3Identify SubproblemsAnswer
1400 cm
Count by edge instead: the height is crossed 8 times, the depth 6 times and the length 2 times, giving 640 + 540 + 220 = 1400 cm.
Takeaway
Ribbon lying flat on a box is the box's own edges in disguise -- including the runs you cannot see.
- Follow one band of the three
- Count the three bands
- Follow the crossing band
- Add everything