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Door 11 · Grade 8

📐 Polygons

A polygon is a closed figure with straight sides

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Types of Polygons and Their Properties

What is a Polygon? A polygon is a closed two-dimensional figure formed by three or more straight line segments. The word comes from Greek: "poly" means many and "gon" means angles.

Classification of Polygons by Number of Sides:

Number of SidesNameExample3TriangleThree-sided figure4QuadrilateralSquare, rectangle5PentagonFive-sided figure6HexagonSix-sided figure7HeptagonSeven-sided figure8OctagonEight-sided figure9NonagonNine-sided figure10DecagonTen-sided figure
Types of Polygons based on Shape:

TypeDescriptionConvex polygonAll interior angles Concave polygonAt least one interior angle > 180°, at least one diagonal lies outsideRegular polygonAll sides equal AND all angles equalIrregular polygonSides and/or angles are not all equal
Key Properties of Polygons:

• A polygon has the same number of sides, vertices, and interior angles
• The sum of exterior angles of any…

Worked example

Problem: Name the polygon with: (a) 6 sides (b) 8 sides (c) 5 sides

  1. - (a) 6 sides → Hexagon
  2. - (b) 8 sides → Octagon
  3. - (c) 5 sides → Pentagon
  4. - **
  5. *Example 2:

Answer: ** (a) Hexagon, (b) Octagon, (c) Pentagon

Interior and Exterior Angles of Polygons

Interior Angles of a Polygon: The angles inside a polygon formed by two adjacent sides are called interior angles.

Formula for Sum of Interior Angles: For a polygon with \(n\) sides:

\[ Sum of interior angles = (n - 2) × 180° \]
Each Interior Angle of a Regular Polygon:

\[ Each interior angle = (n - 2) × 180°/n \]
Exterior Angles of a Polygon: An exterior angle is formed by extending one side of a polygon. It is supplementary to the adjacent interior angle.

Important Properties:

• The sum of exterior angles of any polygon (taken one at each vertex) is 360°
• Each exterior angle of a regular polygon = \(360°/n\)
• Interior angle + Exterior angle = \(180°\) (they form a linear pair)

Relationship Table:

Polygon (n sides)Sum of Interior AnglesEach Interior Angle (Regular)Each Exterior Angle (Regular)Triangle (3)180°60°120°Quadrilateral (4)360°90°90°Pentagon…

Worked example

Problem: Find the sum of interior angles of a hexagon.

  1. - Hexagon has \(n = 6\) sides
  2. - Sum = \((n - 2) × 180° = (6 - 2) × 180° = 4 × 180° = 720°\)
  3. - **
  4. *Example 2:
  5. Find the measure of each interior angle of a regular octagon.

Answer: ** 720°

Regular Polygons and Tessellations

What is a Regular Polygon? A regular polygon is a polygon that is both equiangular (all angles equal) and equilateral (all sides equal).

Properties of Regular Polygons:

• All sides have the same length
• All interior angles have the same measure
• All exterior angles have the same measure
• Regular polygons are always convex
• They have rotational symmetry and reflection symmetry

What is a Tessellation? A tessellation (or tiling) is a repeating pattern of shapes that covers a plane completely with no gaps and no overlaps.

Which Regular Polygons Tessellate? Only three regular polygons tessellate by themselves:

Regular PolygonInterior AngleFits around point?Tessellates?Equilateral triangle60°6 × 60° = 360°YesSquare90°4 × 90° = 360°YesRegular hexagon120°3 × 120° = 360°Yes
Why others don't tessellate:

• Pentagon (108°): no whole number × 108° = 360°
• Octagon…

Worked example

Problem: Find the measure of each exterior angle of a regular decagon.

  1. - Decagon has \(n = 10\) sides
  2. - Each exterior angle = \(360°/n = 360°/10 = 36°\)
  3. - **
  4. *Example 2:
  5. Can a regular pentagon tessellate by itself?

Answer: ** 36°