Polygons are shapes with straight sides and angles. They come in various forms, from triangles to decagons and beyond. Understanding their properties helps us measure and classify them accurately.

Key aspects of polygons include perimeter, interior and exterior angles, and classification based on sides and angles. Circles, though not polygons, share some similar concepts like circumference and area calculations.

Polygons

Perimeter and circumference calculations

• Perimeter of a polygon
  • Represents the total distance around the polygon by adding the lengths of all sides
  • Calculated using the formula $P = s_1 + s_2 + ... + s_n$, where $s_i$ is the length of each side (square, pentagon)
• Circumference of a circle
  • Measures the distance around the circle, similar to the perimeter of a polygon
  • Determined using the formula $C = 2\pi r$, where $r$ is the radius, or $C = \pi d$, where $d$ is the diameter (coin, pizza)
• Measuring lengths
  • Involves using measuring tools like rulers or tape measures to find side lengths or radius/diameter
  • Requires converting all measurements to the same unit (inches, centimeters) before calculating perimeter or circumference

Interior and exterior angle measures

• Interior angles
  • Formed inside the polygon where two adjacent sides meet
  • Sum of all interior angles calculated using $(n - 2) \times 180°$, where $n$ is the number of sides (triangle: $180°$, square: $360°$)
  • Each interior angle in a regular polygon measures $\frac{(n - 2) \times 180°}{n}$ (regular pentagon: $108°$)
• Exterior angles
  • Created by extending a side of the polygon outward
  • Always sum to $360°$ for any polygon, regardless of the number of sides
  • Measure of each exterior angle in a regular polygon is $\frac{360°}{n}$ (regular hexagon: $60°$)
• Relationship between interior and exterior angles
  • Adjacent interior and exterior angles form a straight line, making them supplementary (sum to $180°$)

Classification of polygons

• Number of sides
  • Polygons classified based on the number of sides they have (triangle: 3, octagon: 8)
  • Names for polygons with 3 to 10 sides: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon
• Angles
  • Acute polygons have all angles measuring less than $90°$ (acute triangle)
  • Obtuse polygons have at least one angle greater than $90°$ (obtuse triangle)
  • Right polygons have at least one angle equal to $90°$ (right triangle, rectangle)
• Sides
  • Equilateral polygons have all sides equal in length (equilateral triangle, square)
  • Equiangular polygons have all angles equal in measure (square, regular pentagon)
  • Regular polygons are both equilateral and equiangular (square, regular hexagon)
• Convexity
  • Convex polygons have all interior angles less than $180°$ (square, regular pentagon)
  • Concave polygons have at least one interior angle greater than $180°$ (star shape, arrow shape)
• Symmetry
  • Line symmetry occurs when a polygon can be divided into two congruent halves by a line (rectangle, isosceles triangle)
  • Rotational symmetry happens when a polygon appears unchanged after rotating it about its center by a certain angle (square: $90°$, $180°$, $270°$, equilateral triangle: $120°$, $240°$)

Basic elements of polygons

• Sides: The line segments that form the boundary of the polygon
• Vertices: The points where two sides of a polygon meet (singular: vertex)
• Angles: Formed by two adjacent sides of the polygon
• Area: The space enclosed within the polygon's boundaries

Circles

• A round shape with all points on its edge equidistant from the center
• Circumference is the distance around the circle (perimeter equivalent)
• Area is calculated using the formula $A = \pi r^2$, where $r$ is the radius