Polygon Angles & Properties
SAT Math — Geometry: The Complete Guide to Polygons
Master every polygon formula the SAT can throw at you — from angle sums to diagonals to similar-polygon ratios.
1. What Is a Polygon?
A polygon is a closed plane figure formed by three or more line segments (called sides) connected end-to-end. Each point where two sides meet is a vertex (plural: vertices).
| Sides | Name | Sides | Name |
|---|---|---|---|
| 3 | Triangle | 8 | Octagon |
| 4 | Quadrilateral | 9 | Nonagon |
| 5 | Pentagon | 10 | Decagon |
| 6 | Hexagon | 12 | Dodecagon |
| 7 | Heptagon | $n$ | $n$-gon |
Interactive visualization — available in the full lesson
Key Vocabulary
Side — line segment forming part of the boundary
Vertex — point where two sides meet ($n$ sides → $n$ vertices)
Diagonal — segment connecting two non-adjacent vertices
Interior angle — angle inside the polygon at a vertex
Exterior angle — angle between one side and the extension of the adjacent side
Perimeter — total length of all sides | Area — enclosed region
SAT Relevance: Polygons appear in area, angle, coordinate-geometry, and data-analysis problems. Knowing the formulas cold saves precious time.
2. Convex vs. Concave
- Convex polygon — every interior angle < 180°; all vertices "point outward"; every diagonal lies inside.
- Concave polygon — at least one interior angle > 180°; at least one vertex "dents inward"; at least one diagonal lies partially outside.
SAT Tip: Almost all SAT polygon problems involve convex polygons. If a problem doesn't specify, assume convex.
3. Regular Polygons
A polygon is regular if it is both:
- Equilateral — all sides equal length
- Equiangular — all angles equal measure
Examples: equilateral triangle, square, regular pentagon, regular hexagon, regular octagon.
Watch out! A rectangle is equiangular but NOT equilateral (unless it's a square). A rhombus is equilateral but NOT equiangular (unless it's a square). Neither is regular on its own.