Problem
Vertex labels on a regular octahedron net
A regular octahedron has its six vertices named. Beside it is a net of that same solid. One triangle of the net is already labelled. Fill in the names on the rest of the net.
Reasoning · Grade 6-1 Regular Polyhedra
Your answer
Watch out Filling names freely fails: opposite vertices can never share a triangle.
How to solve
Strategy Create a Physical Representation — Guessing which points meet by eye is exactly how this kind of question goes wrong, so instead I fold the net one triangle at a time. The trick is that I never need to picture the whole solid at once. Two triangles that touch along an edge of the net are two faces that touch along that edge of the solid — and on an octahedron every edge has exactly two faces, so once I know one of them, the other is forced. The net's 8 triangles form a chain, so I start from the one triangle whose three names are already printed and walk along the chain, and at every step exactly one new name gets decided. First I list the octahedron's 8 faces so I have something to look each edge up in.
1STEP 1
List the 8 faces of the octahedron
List the solid's eight faces.
upper: ABC, ACD, ADE, AEB lower: FBC, FCD, FDE, FEB
Reading the eight faces straight off the picture turns a hard folding question into a lookup: given an edge, which face is on the other side of it? A written list answers that instantly, with no mental rotation at all.
4.G.A.1Make A Systematic List2STEP 2
Start from the triangle that is already named
Start from the labelled triangle.
A net is just the solid's surface cut open and flattened, so a triangle drawn on the net IS a face of the solid — the printed names hand you one face for free, and that is enough to start.
6.G.A.4Create A Physical Representation3STEP 3
Fold once: the first strip triangle, so L1 = F
Each shared edge has only one other face, fixing the name.
edge BC ⊂ {ABC, FBC} → L₁ = F
This is the one move the whole solution repeats: an edge has exactly two faces, so knowing one face and the edge names the other face with no choice left.
6.G.A.4Create A Physical Representation4STEP 4
Keep walking: L2 = D, then U3 = A
Walking on, fill the lower row.
edge FC → FCD, L₂ = D edge CD → ACD, U₃ = A
Each step reuses two names you already have and adds one new one, so there is never more than one thing to decide and never any guessing.
6.G.A.4Create A Physical Representation5STEP 5
Finish the strip: L3 = E, U4 = B, L4 = F
Finish the strip and fill the upper row.
DA → ADE, L₃ = E AE → AEB, U₄ = B EB → FEB, L₄ = F
The names run round the middle square in order — C, D, then E, then back to B — because walking along the strip is walking once round the equator of the solid.
6.G.A.4Create A Physical Representation6STEP 6
The last triangle: BOT = D
The last triangle's apex is forced too.
edge EF ⊂ {FEB, FDE} → BOT = D
The eighth triangle is the last face left over from the list of eight, so it had to be FDE even before the edge was checked — a nice confirmation that nothing was skipped.
6.G.A.4Create A Physical Representation7STEP 7
Check every name collects a full 240 degrees
Each name collects four triangles.
4 × 60° = 240° at every vertex; 8 × 3 = 24 = 6 × 4
This check catches almost every slip. If a name had picked up only 3 triangles or as many as 5, the folded paper would either gape open or overlap at that corner, so a wrong labelling shows up as a wrong angle total.
4.MD.C.7Make A Systematic ListEvery vertex name must collect a full 240 degrees of triangle corners, which checks the whole labelling at once.
Why?
Four equilateral triangles meet at each vertex of an octahedron, and their corner angles sit side by side around that point.
🧱Angle measure addsA ray drawn inside an angle cuts it into two smaller angles whose measures add back to the original, just as lengths do.
Why?
Those angles fall short of a full turn, and by exactly the same amount at every vertex, so a name collecting the wrong total must be wrong.
🧱Angles around a point make 360 degreesAngles that fit side by side all the way around one point fill exactly one full turn, 360 degrees.
8STEP 8
Check that opposite vertices never touch
No triangle holds two opposite vertices.
A second, completely different check: the first one counted angles, this one looks at which names are allowed to sit in the same triangle. Passing both makes a mistaken labelling very hard to hide.
4.G.A.1Visualize Spatial RelationshipsAnswer
F, D, E, F / A, B / D
8 × 3 = 24 = 6 × 4
The ten net points carry six different names, which is right because the octahedron has six vertices, and the counts fit: four names appear twice and two appear once, giving 4 x 2 + 2 x 1 = 10 points. The eight labelled triangles turn out to be exactly the eight faces ABC, ACD, ADE, AEB, FBC, FCD, FDE, FEB, each used once, so no face is repeated or missing. Every name collects exactly 4 triangles, 240 degrees, which is what a vertex of a regular octahedron needs; and the labelled edges come out as 12 different pairs, each shared by exactly 2 faces, matching the octahedron's 12 edges. Finally, no triangle contains an opposite pair (A with F, B with D, C with E), as it must not. Folding a paper copy and reading the names off the finished solid confirms it.
Takeaway
Every edge of a solid has exactly two faces, so once one triangle of the net is named you can walk along the net naming one new point at a time — no guessing, and the 240 degrees at each corner proves you got it right.
- List the 8 faces of the octahedron
- Start from the triangle that is already named
- Fold once: the first strip triangle, so L1 = F
- Keep walking: L2 = D, then U3 = A
- Finish the strip: L3 = E, U4 = B, L4 = F
- The last triangle: BOT = D
- Check every name collects a full 240 degrees
- Check that opposite vertices never touch