Problem
Longest and shortest perimeter of a net
A prism measures 3 cm by 1 cm by 2 cm. Part (1) prints three different nets of it. Part (2) asks for a new net of the same prism. Find the three perimeters and the shortest possible one.
Reasoning · Grade 5-2 Rectangular Prisms
Your answer
Watch out A fold seems to cost its own length, but both sides vanish — it is twice.
How to solve
Strategy Identify Subproblems — Walking round the outline of a stepped net counting centimetres works, but it is slow and it is very easy to skip a step, and for part (2) it tells you nothing at all about which net is best. So I break the perimeter into two easier pieces instead: start from all six faces lying separately, whose perimeters are easy to add, and then subtract what is lost each time two faces are joined. That turns the whole question into a much smaller one — which 5 edges are left as folds — and once the problem is phrased that way, making the perimeter short simply means choosing the longest edges to be the folds. I then list the 12 edges by length to see which choices are actually allowed, and check the final answer by walking the boundary of a drawn net and by folding a paper model.
1STEP 1
Add up the perimeters of the six separate faces
The six separate faces total 48 cm.
2 × 10 + 2 × 8 + 2 × 6 = 20 + 16 + 12 = 48 cm
Adding up the six rectangles one at a time is ordinary Grade 3 perimeter work, and it gives a fixed starting number that the shape of the net can never change.
3.MD.D.8Identify Subproblems2STEP 2
See what each fold line costs
Each fold removes twice its length.
perimeter of net = 48 - 2 × (total length of the fold lines)
Two edges disappear inside the net at every join, not one, and that doubling is the whole reason long folds are worth so much.
6.G.A.4Identify SubproblemsEach fold line removes two edge lengths from the net's perimeter, because it hides one side of each of two faces.
Why?
A fold line belongs to two faces at once and is counted once from each, so joining costs two sides of boundary.
🧱Counted twice, so halveIf every item got counted from both of its two ends, the total is exactly double the truth, so halving fixes it.
Why?
The net's perimeter is all the faces' sides put together minus the ones the folds have hidden.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Check that every net has exactly 5 fold lines
Every net has 5 folds.
6 faces → 6 - 1 = 5 folds, 12 - 5 = 7 cuts
Cut a real cereal box open and count: it always takes exactly the same number of cuts, because 6 pieces need 5 links to stay in one chain, no matter how the chain is shaped.
6.G.A.4Create A Physical Representation4STEP 4
First printed net: read its five folds
The first net measures 28 cm around.
2+1+3+3+1 = 10, 48 - 2 × 10 = 28 cm
The dashes are drawn for you, so this is just reading five numbers off the picture and adding them — far safer than counting thirty-odd steps around a staircase edge.
4.OA.A.3Make A Systematic List5STEP 5
Second and third printed nets
The others measure 24 cm and 26 cm.
3+3+3+2+1 = 12 → 48 - 24 = 24 cm; 3+2+2+3+1 = 11 → 48 - 22 = 26 cm
Three nets of one prism giving three different perimeters is exactly the point of the question: the faces are fixed, but how much edge is hidden inside is not.
4.OA.A.3Make A Systematic List6STEP 6
Turn 'shortest perimeter' into a question about edges
A short perimeter needs the longest total fold.
perimeter smallest ⇔ total fold length largest
Swapping a hard drawing question for an easy choosing question is the real move here, and it works because the 48 cm of face perimeter is the same for every net.
6.G.A.4Identify Subproblems7STEP 7
List the 12 edges and see which five can be folds
The folds can total at most 13 cm.
3 + 3 + 3 + 2 + 2 = 13 cm is the greatest possible total fold length
Sorting the twelve edges into three groups of four shows immediately which faces each length can possibly join, and the no-ring rule is what stops the greedy choice of all four long edges.
6.G.A.4Make A Systematic List8STEP 8
Work out the shortest perimeter and draw such a net
So the shortest perimeter is 22 cm.
48 - 2 × 13 = 48 - 26 = 22 cm
Wrapping the four biggest faces around the prism in one band is what puts three whole 3 cm edges inside the net, and the two small faces are then tucked on by their longest available sides.
6.G.A.4Draw A Diagram9STEP 9
Check the drawing by walking round it
Walking round the drawn net gives 22 cm.
12 + 10 = 22 cm
This particular outline has no notch pointing sideways, so its boundary can be slid out to the 6 cm by 5 cm rectangle around it without changing length — a check worth doing but not a rule to trust on every net, since a net with a dent in it has a longer boundary than the rectangle around it.
3.MD.D.8Create A Physical RepresentationAnswer
28, 24, 26 / 22 cm
48 − 2 × 13 = 22
Every answer is an even number of centimetres, which it must be, since the perimeter is 48 minus twice something. All four values sit between the sensible extremes: the folds can total at most 13 cm, so no net can beat 22 cm, and the three printed nets at 28, 24 and 26 cm are all longer than that, as they have to be. The answer of 22 cm was reached two independent ways — once by the rule 48 minus twice 13, and once by counting round the drawn outline, 6 plus 5 doubled — and both give 22. It is also worth checking the totals are believable: the whole boundary of six loose faces is 48 cm, and the best net hides more than half of that inside as folds, which matches the picture of a tightly wrapped net with very little edge showing.
Takeaway
A net's border is always 48 cm minus double the length of its fold lines, so keeping the longest edges as folds is what makes the border shortest!
- Add up the perimeters of the six separate faces
- See what each fold line costs
- Check that every net has exactly 5 fold lines
- First printed net: read its five folds
- Second and third printed nets
- Turn 'shortest perimeter' into a question about edges
- List the 12 edges and see which five can be folds
- Work out the shortest perimeter and draw such a net
- Check the drawing by walking round it