Polygons consist of line segments connected only at their endpoints. Many types of polygons exist, with varying side and angle measurements. Using formulas, we can find the sum of the interior angles of a polygon.
Polygon: Any closed planar figure made entirely of line segments that only intersect at the endpoints.
Side: The line segment making up the polygon.
Vertex (plural, vertices): The point where the segments intersect.
Convex Polygon: A polygon where all the interior angles are less than 180°.
Interior Angle: The angle inside of a closed figure with straight sides.
Concave Polygon: A polygon where at least one interior angle is greater than 180°.
Diagonal: A line segment that connects two non-consecutive sides of a polygon.
Exterior Angle: The angle formed by one side of a polygon and the extension of the adjacent side.
Equilateral Polygon: All sides of the polygon are congruent.
Equiangular Polygon: All interior angles of the polygon are congruent.
Regular Polygon: A polygon that is equilateral and equiangular.
In a convex polygon, no section points inward.
A side connects consecutive vertices, while a diagonal connects nonconsecutive vertices.
A concave polygon has a section that points inward toward the interior of the shape.
A concave polygon has a section that caves in.
Theorem: For any convex polygon, the sum of the interior angles is (n - 2) • 180°
Sum of interior angle measures = (3 - 2) • 180° = 180°
To remember the formula for the sum of the interior angles of a polygon, draw the polygon and cut it into triangles by drawing all the diagonals from one vertex.
For any equiangular polygon, the measure of each angle is \(\frac{(n - 2) . 180^{\circ}}{n}\).
Since regular polygons are equiangular and equilateral, this formula can be used for regular polygons as well.
The sum of the exterior angles for a convex polygon does not depend on the number of sides.
Just like triangles, polygons have two sets of exterior angles.
Polygons can be classified by the number of sides. An n-gon is a polygon with n sides. The table below lists some of the common polygons that have special names.
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Heptagon
7
Octagon
8
Nonagon
9
Decagon
10
Undecagon
or hendecagon
11
Dodecagon
12