Polygon triangle fan
Count noncrossing triangles and justify a convex polygon’s interior-angle sum.
Adult planning suggestion: about 9–12 · Shape & spatial thinking
- Players
- 1–4 cooperatively
- Time
- 15–25 minutes
- Materials
- Convex polygons with 3–8 numbered vertices
Prepare
Use a convex pentagon with vertices 1–5 in boundary order. Convex means every interior angle is below 180 degrees. The supplied drawings are convex; do not apply this exact fan procedure to an arbitrary concave outline.
Play
From vertex 1 draw diagonals to vertices 3 and 4. Do not count boundary sides 1–2 or 1–5 as diagonals. The two diagonals lie inside and divide the pentagon into three triangles without overlap.
Worked example
Each triangle has angle sum 180 degrees. The triangle angles at the original vertices together fill the pentagon’s angles, so its sum is 3×180=540 degrees. This gives a sum, not the value of every individual angle.
Finish and check
Repeat on convex polygons with 3,4,6,7,8 vertices. From vertex 1 join vertices 3 through n−1. The fan has n−3 diagonals and n−2 triangles. Fill both counts and 180(n−2) in the table.
Adaptation and limits
A partner checks the drawn diagonals and triangle count before the multiplication. Finish when the table and drawings agree. Only for a regular polygon may the sum be divided by n to get every equal angle; an irregular pentagon need not have five 108-degree angles.
Purpose, observation and provenance
Within this activity: Count noncrossing triangles and justify a convex polygon’s interior-angle sum.
Ask learners to distinguish boundary sides from diagonals and show which triangles jointly fill each original vertex angle.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
OpenStax: Polygons — Supports convex polygon decomposition and angle sums; our numbered drawings and worksheets are original.
Secondary practice: Cooperation and explaining a check.
For a class of 30, use ten groups of three. Prepare one player-material set per group and keep the host key with the adult. Rotate operator, recorder and checker after each worked example; a reader or scribe may help without choosing answers. Avoid public ranking or diagnostic labels.
Age bands are planning suggestions. Background references may address older learners and do not validate suitability for this age band. Goals describe observable actions, not measured long-term gains or institutional endorsement.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Rules to keep beside the materials
- Use a convex pentagon with vertices 1–5 in boundary order. Convex means every interior angle is below 180 degrees. The supplied drawings are convex; do not apply this exact fan procedure to an arbitrary concave outline.
- From vertex 1 draw diagonals to vertices 3 and 4. Do not count boundary sides 1–2 or 1–5 as diagonals. The two diagonals lie inside and divide the pentagon into three triangles without overlap.
- Each triangle has angle sum 180 degrees. The triangle angles at the original vertices together fill the pentagon’s angles, so its sum is 3×180=540 degrees. This gives a sum, not the value of every individual angle.
- Repeat on convex polygons with 3,4,6,7,8 vertices. From vertex 1 join vertices 3 through n−1. The fan has n−3 diagonals and n−2 triangles. Fill both counts and 180(n−2) in the table.
- A partner checks the drawn diagonals and triangle count before the multiplication. Finish when the table and drawings agree. Only for a regular polygon may the sum be divided by n to get every equal angle; an irregular pentagon need not have five 108-degree angles.
Polygon triangle fan — play materials
| n | Diagonals | Triangles | Angle sum ° |
|---|---|---|---|
| 3 | |||
| 4 | |||
| 5 | |||
| 6 | |||
| 7 | |||
| 8 |
Host check — keep private until finished
n 3–8: triangles 1–6; diagonals 0–5; angle sums 180,360,540,720,900,1080.
These are model answers or legal examples, not records of an experiment or learner outcomes.
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