Geometry

Polygons

Big Picture

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.

Key Terms

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.

Polygons

  • In a polygon, the sides can NEVER be curved. Line segments form the sides of the polygon.
  • The polygon must also be closed.
  • To name a polygon, the vertices are listed in consecutive order.
  • A polygon lies within one plane.

Convex and Concave Polygons

In a convex polygon, no section points inward.

A side connects consecutive vertices, while a diagonal connects nonconsecutive vertices.

    convex polygon

      A concave polygon has a section that points inward toward the interior of the shape.

      • Concave polygons have at least one diagonal outside the figure (the exterior line drawn in the figure below)

      A concave polygon has a section that caves in.

        Concave polygons

        Geomerty

        Polygons cont.

        Angles in Polygons

        Interior Angles in Convex Polygons

        Interior Angles in Convex Polygons
        • n = number of sides of a polygon
        • Number of interior angles in a polygon = number of sides = n

        Theorem: For any convex polygon, the sum of the interior angles is (n - 2) • 180°

        • Recall that the interior angles of a triangle adds up to 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.

        Interior Angles
        • Count the number of triangles and multiply that number by 180°.

        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.

        Exterior Angles in Convex Polygons

        The sum of the exterior angles for a convex polygon does not depend on the number of sides.

        • The sum of the exterior angles of a convex polygon is always 360°.

        Just like triangles, polygons have two sets of exterior angles.

        Exterior Angles in Convex Polygons

        Geometry

        Polygons cont.

        Classifying Polygons

        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.

        Polygon Name

        Number
        of Sides

        Regular Polygon

        Triangle

        3

        Triangle

        Quadrilateral

        4

        Quadrilateral

        Pentagon

        5

        Pentagon

        Hexagon

        6

        Hexagon

        Heptagon

        7

        Heptagon

        Polygon Name

        Number
        of Sides

        Regular Polygon

        Octagon

        8

        Octagon

        Nonagon

        9

        Nonagon

        Decagon

        10

        Decagon

        Undecagon  
        or hendecagon

        11

        hendecagon

        Dodecagon

        12

        Dodecagon

        Notes