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Math101Polygons
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 $360^\circ$.
