Geometry: Polygons and Angles

Polygon Basics

• A polygon is a simple closed plane figure formed by three or more line segments joined end to end, no two of which in succession are collinear.
• The line segments forming the polygon are called sides.
• The common end point of any two sides is called a vertex of the polygon.
• The angles formed inside the polygon are called interior angles.

Types of Polygons

• A convex polygon is a polygon where the measure of each interior angle is less than 180°.
• A concave polygon is a polygon where there is at least one interior angle whose measure is more than 180 degrees.

Angles in Polygons

• The sum of the measure of interior angles of an 𝑛-sided polygon is determined by (𝑛 − 2) × 180°.
• The sum of the measure of exterior angles of a polygon is 360°.
• At each vertex of a polygon, the external and internal angle add up to 180°.

Regular Polygons

• A regular polygon is a convex polygon with all sides equal and all interior angles equal.
• The measure of each interior angle of a regular 𝑛-sided polygon is (𝑛−2)×180°/𝑛.
• The measure of each exterior angle of a regular 𝑛-sided polygon is 360°/𝑛.
• The measure of each central angle of a regular 𝑛-sided polygon is 360°/𝑛.

Symmetry

• If two parts of a figure are identical after folding through the line of symmetry, then it is said to be symmetric.
• A symmetrical figure has at least one line of symmetry.
• An 𝑛-sided regular polygon has 𝑛 lines of symmetry.

Inscribed and Circumscribed Polygons

• An inscribed polygon is a polygon in which all vertices lie on a circle.
• A circumscribed polygon is a polygon in which each side touches a circle.

Apothem and Formulas

• An apothem of a regular polygon is a perpendicular segment from a midpoint of a side of a regular polygon to the centre of the circle circumscribed about the polygon.
• Formulas for the length of side 𝑠, apothem 𝑎, perimeter 𝑃, and area 𝐴 of the regular polygon with 𝑛-sides and radius 𝑟 are given by:
  • 𝑠 = 2𝑟 sin (180°/𝑛)
  • 𝑎 = 𝑟 cos (180°/𝑛)
  • 𝑃 = 2𝑛𝑟 sin (180°/𝑛)
  • 𝐴 = (1/2)𝑎𝑃