Problem
Edges of a solid from its net
A net of 3 squares and 4 triangles is drawn. Six dashed lines are fold lines. The rest make up the outer boundary. Count the edges of the solid it folds into.
Reasoning · Grade 6-1 Using Nets
Your answer
Watch out Counting boundary lines overcounts: folding makes two lines into one edge.
How to solve
Strategy Make a Systematic List — Folding a 7-faced solid in your head and then counting its edges one by one is exactly where an answer goes wrong, so I count on the flat picture instead, where every line is visible. I split the edges into two kinds — the ones that come from a fold and the ones that come from taping two boundary lines together — and count each kind separately. To count the boundary lines without trusting my eyes (two of them lie along one straight line and look like a single line), I count the sides of every face first and subtract the ones used by the folds. Then I check the total against a picture of the actual solid.
1STEP 1
Count the sides of all the faces
All the faces' sides total 24.
3 × 4 + 3 × 3 + 3 = 12 + 9 + 3 = 24
Counting sides face by face is safe multiplication and addition, and unlike counting lines in the drawing it cannot be fooled by two sides that happen to lie along one straight line.
4.OA.A.3Make A Systematic List2STEP 2
See what a dashed fold line does
The six folds use 12 of those sides.
6 × 2 = 12 sides used, 6 edges
A crease in a folded piece of paper is one edge, and the paper on the two sides of it is the two faces — that is exactly what a dashed line in a net means.
6.G.A.4Visualize Spatial RelationshipsA dashed fold line is one edge of the solid, and it uses up one side from each of the two faces it joins.
Why?
The fold belongs to two faces at once and is counted once from each, so two drawn sides become one real edge.
🧱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?
Each fold line joins exactly one pair of faces, so folds and joined pairs match up one for one with none left over.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Count the boundary lines
That leaves 12 boundary lines.
24 - 12 = 12
Subtracting the sides the folds have already claimed leaves exactly the sides that are still free, and free sides are what the outer boundary is made of.
4.OA.A.3Draw A Diagram4STEP 4
Turn the boundary lines into edges
They pair up into 6 edges.
12 ÷ 2 = 6
Cutting one edge of a closed solid opens it into two boundary lines, so running the film backwards, two boundary lines always close back into one edge.
6.G.A.4Visualize Spatial Relationships5STEP 5
Check the pairing by length
The lengths pair up correctly.
8 ÷ 2 = 4, 4 ÷ 2 = 2, 4 + 2 = 6
Two boundary lines can only be taped together if they are the same length, so sorting them by length is a free check that the pairing is possible at all.
4.OA.A.3Make A Systematic List6STEP 6
Add the two kinds of edge
Adding the 6 folds gives 12 edges.
6 + 6 = 12
Splitting a count into two kinds that never overlap and never leave anything out means the two counts can simply be added.
4.OA.A.3Identify Subproblems7STEP 7
Check against the solid itself
Picturing the solid also gives 12.
12 - 3 + 3 = 12
Naming the solid turns the answer into something you can picture and recount a completely different way, which is worth far more than repeating the same count twice.
6.G.A.4Visualize Spatial RelationshipsAnswer
12 edges
6 + 6 = 12
The number is a count of edges, so it must be a whole number, and it is: the 12 boundary lines split evenly into 6 pairs with the lengths matching (8 short lines, 4 diagonal lines), which they would not do if a line had been miscounted. The 6 fold lines join all 7 faces into one connected piece with no fold to spare, which is right — a net of 7 faces always needs exactly 7 - 1 = 6 folds. And the answer agrees with Euler's rule for a solid like this one: with 7 faces and 12 edges the solid should have 12 - 7 + 2 = 7 corners, and the cube with a corner sliced off has exactly 7 corners (the cube's 8 corners, less the one cut away).
Takeaway
Count on the flat net, not in your head: a dashed crease is one edge, and two solid boundary lines always tape together into one more.
- Count the sides of all the faces
- See what a dashed fold line does
- Count the boundary lines
- Turn the boundary lines into edges
- Check the pairing by length
- Add the two kinds of edge
- Check against the solid itself