Polygons
Overview
This lesson names polygons by their number of sides and studies their angles. Drawing the diagonals from one vertex splits a convex polygon into (n − 2) triangles, which gives the interior angle sum S = (n − 2) · 180°. The exterior angles, one at each vertex, always add to 360°, and in a regular polygon — convex with all sides and all angles congruent — those totals divide evenly among the angles.
Vocabulary
- Polygon
- A multi-sided closed figure whose sides are segments that intersect only at their endpoints.
- Convex Polygon
- A polygon in which no line containing a side of the polygon also contains a point in the interior of the polygon. Simply: all vertices point out.
- Concave Polygon
- A polygon that is not convex. Simply: at least one vertex points in.
- Regular Polygon
- A convex polygon in which all sides and all angles are congruent.
- Exterior Angle of a Polygon
- The angle formed at a vertex between one side of the polygon and the extension of the adjacent side. An exterior angle and its adjacent interior angle are supplementary.
- n-gon
- A polygon with n sides. Special names: 3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 7 heptagon, 8 octagon, 9 nonagon, 10 decagon, 11 hendecagon, 12 dodecagon.
Main Ideas
- Polygons are classified by their number of sides, and the number of sides equals the number of interior angles and the number of exterior angles.
- The sum of the interior angles of a convex polygon is S = (n − 2) · 180°, because diagonals from one vertex form (n − 2) triangles.
- The sum of the exterior angles of a convex polygon, one angle at each vertex, is always 360°, no matter how many sides the polygon has.
- An interior angle and its adjacent exterior angle are supplementary — they add to 180°.
- In a regular polygon, each interior angle measures (n − 2) · 180° ÷ n, and each exterior angle measures 360° ÷ n.
- To find the number of sides, work the formulas backward: divide an interior sum by 180° and add 2, or divide 360° by the measure of one exterior angle of a regular polygon.
Lesson Notes
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Lesson Video
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Practice Problems
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