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Math101learn.math101.caPolygons
A polygon is a closed plane figure made of finitely many straight segments meeting endpoint to endpoint, with standard simple polygons having noncrossing sides.
Polygon formulas organize shape classification, tiling, design, construction, and proofs by reducing many-sided figures to triangles and turns.
Intuition and core definition
A polygon is a closed plane figure made of finitely many straight segments meeting endpoint to endpoint, with standard simple polygons having noncrossing sides. An $n$-gon has $n$ sides and vertices. Regular means both equilateral and equiangular.
Notation, language, and conditions
Interior-angle sum is $(n-2)180^\circ$ for a simple $n$-gon. A regular polygon has each interior angle $(n-2)180^\circ/n$ and each exterior turning angle $360^\circ/n$. The number of diagonals is $n(n-3)/2$.
Why this idea matters
Polygon properties connect side count, angle sums, regularity, and diagonal structure in closed straight-sided figures.
A dependable method
- Confirm the figure is closed and built from line segments.
- Count sides or vertices and determine whether it is simple, convex/concave, and regular.
- Choose the appropriate total or per-angle formula.
- Substitute $n\ge3$ and retain degree units.
- Check interior and exterior angles are supplementary at each vertex and exterior turns total $360^\circ$.
Worked example
Representations and interpretation
Drawing diagonals from one vertex partitions a convex $n$-gon into $n-2$ triangles, deriving the interior-sum formula. Walking around the boundary accumulates one full exterior turn.
Reasoning about variations
The interior-sum formula still holds for simple concave polygons when interior angles are interpreted correctly. Self-intersecting star polygons require different conventions and should not be inserted without checking assumptions.
Common mistakes
How to check your work
- Triangulate from one vertex and count $n-2$ triangles.
- Add regular interior and exterior angles to $180^\circ$.
- Use a full-turn check on exterior angles.
Practice
- Find each interior angle of a regular hexagon.
- How many diagonals does a decagon have?
- Is every equilateral polygon regular?
Answers and brief solutions
Show answers
- $120^\circ$ The sum is $720^\circ$ and $720/6=120$.
- $35$ $10(10-3)/2=35$.
- No It must also be equiangular.
Synthesis and transfer
Triangulating a many-sided floor plan from one vertex explains its interior-angle sum, while a concave example tests whether the decomposition is interpreted carefully.
Drawing diagonals from one vertex of a simple convex $n$-gon partitions it into $n-2$ triangles, giving interior-angle sum $(n-2)180^\circ$. For a regular polygon, division by $n$ yields each interior angle, while each exterior turning angle is $360^\circ/n$. A concave polygon still has the same interior sum, although a careless fan drawn outside the figure can obscure the decomposition. The exterior-turn total remains one full rotation for any consistently traversed simple polygon. Checking small cases—triangle and quadrilateral—anchors the general formula and catches an incorrect $n-1$ or $n$ factor.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find each interior angle of a regular hexagon.
- The sum is $720^\circ$ and $720/6=120$.
End of lesson
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