Problem
Shortest string tied around a box
A box measures 6 cm across, 4 cm deep and 10 cm tall. A string is tied all the way round in two different ways. Each way crosses three named edges. Find the shortest length for each way.
Reasoning · Grade 6-1 Using Nets
Your answer
Watch out Measuring face by face is messy: unroll and draw one straight line.
How to solve
Strategy Draw a Diagram — On the box the string bends at every edge, and a bent line is impossible to measure. So for each method I first work out which faces the string crosses, then unroll just those faces flat, like peeling a paper band off the box. On the flat picture the string has to join two points and the shortest line between two points is the straight one, so the answer can simply be read off the drawing. Peeling a real band of paper off a real box is the check that the unrolled picture is the right one.
1STEP 1
Write down every edge length
Note the edges 6, 4 and 10.
width = 6 cm, depth = 4 cm, height = 10 cm
Three numbers describe the whole box, because parallel edges of a box are equal — so the one 6, one 4 and one 10 in the picture give all twelve edge lengths.
6.G.A.4Identify Subproblems2STEP 2
Method 1: find which faces the string crosses
Find the four faces the first way crosses.
Naming the edges the string crosses is the same as naming the faces between them, because two named edges next to each other always have exactly one face in between.
6.G.A.4Visualize Spatial Relationships3STEP 3
Method 1: unroll the band flat
Unrolled, that band is 28 long.
4 + 10 + 4 + 10 = 28
Unrolling never stretches or squashes the paper, so every length on the box is still the same length on the flat picture — which is exactly why the flat picture can be used to measure the string.
6.G.A.4Draw A DiagramUnrolling the band of faces flat lays the string out without stretching it, so its length is unchanged.
Why?
Unrolling turns each face over its fold without bending or stretching it, so every length on the surface is carried across exactly.
🧱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).
Why?
The string is its pieces on each face laid end to end, so once the faces sit flat the pieces line up into one straight run.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
4STEP 4
Method 1: draw the string straight
Drawn straight, the string is 28 cm.
28 cm
A straight line is the shortest path between two points on flat paper — that one fact is what unrolling was for, and it turns the tightest string into a ruler line.
3.MD.D.8Draw A Diagram5STEP 5
Method 2: find the faces and unroll them
The second band measures 20.
6 + 4 + 6 + 4 = 20
The same peeling trick works in any direction round the box; only which measurements get laid end to end changes.
6.G.A.4Draw A Diagram6STEP 6
Method 2: read off the length
So the second string is 20 cm.
20 cm
Once the picture is flat the answer is just the width of the strip, so no hard measuring is needed at all.
3.MD.D.8Draw A Diagram7STEP 7
Check the two answers another way
Reading them as rectangle perimeters agrees.
2 × (4 + 10) = 28, 2 × (6 + 4) = 20
A tight band round a box lies flat on each face and turns only at the edges, so its length is the perimeter of the rectangle it wraps — a familiar formula giving the same number.
4.MD.A.3Identify SubproblemsAnswer
28, 20 cm
2 × (4 + 10) = 28, 2 × (6 + 4) = 20
Both answers are lengths in centimetres, and both are longer than the longest edge of the box (10 cm), as any string that goes all the way round must be. Method 1 needs more string than Method 2, which fits the picture: Method 1's loop climbs the 10 cm height twice, while Method 2's loop only goes round the 6 cm by 4 cm bottom shape. As a size check, a loose loop taken along the edges of the box would measure 4 + 10 + 4 + 10 = 28 cm and 6 + 4 + 6 + 4 = 20 cm as well, so the tightest string is exactly as long as the edge route and no shorter — which makes sense, because the unrolled string is already straight and cannot be pulled any tighter.
Takeaway
Unroll the faces the string touches and the bent string becomes a straight line you can just measure.
- Write down every edge length
- Method 1: find which faces the string crosses
- Method 1: unroll the band flat
- Method 1: draw the string straight
- Method 2: find the faces and unroll them
- Method 2: read off the length
- Check the two answers another way