{"id":9909,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9909"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"introduction-polygons","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/introduction-polygons\/","title":{"rendered":"Introduction, Polygons"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Understanding Quadrilateral<\/strong><\/h2>\n

Chapter: <\/strong>Introduction to Polygon<\/strong><\/h3>\n

Reference: – What is a Polygon, Types of Polygons (Regular and Irregular), Convex and Concave Polygons, Classification by Number of Sides, Triangle, Quadrilateral, Pentagon, Hexagon, etc., Interior and Exterior Angles, Sum of Interior Angles of a Polygon, Sum of Exterior Angles of a Polygon, Diagonals of a Polygon, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • What is a Polygon<\/em><\/li>\n
  • How to Classify Polygons by Number of Sides<\/em><\/li>\n
  • Difference Between Convex and Concave Polygons<\/em><\/li>\n
  • Difference Between Regular and Irregular Polygons<\/em><\/li>\n
  • Calculate Sum of Interior Angles of a Polygon<\/em><\/li>\n<\/ul>\n

    Introduction to Polygons<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    A polygon is a closed two-dimensional figure formed by three or more straight line segments. The line segments are called sides, and the points where two sides meet are called vertices (singular: vertex). The word "polygon" comes from Greek: poly (many) and gon (angles).<\/p>\n

    When we study polygons, we essentially ask:<\/p>\n

    "What shape is formed by connecting straight line segments in a closed loop?"<\/p>\n

    Once we understand polygons, we can classify them, calculate their angles, and explore their properties.<\/p>\n

    Importance of Polygons<\/u><\/strong><\/p>\n

      \n
    • Found everywhere in nature and design (honeycombs, tiles, buildings)<\/li>\n
    • Foundation for understanding more complex geometric shapes<\/li>\n
    • Used in computer graphics, architecture, and engineering<\/li>\n
    • Helps develop spatial reasoning and problem-solving skills<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      A triangle is a polygon with 3 sides. A square is a polygon with 4 sides. A pentagon is a polygon with 5 sides. A circle is NOT a polygon because it has curved sides, not straight-line segments.<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Parts of a Polygon<\/strong><\/p>\n

      Sides: The straight-line segments that form the polygon<\/p>\n

      Vertices: The points where two sides meet (corners)<\/p>\n

      Diagonals: Line segments joining two non-adjacent vertices<\/p>\n

      Interior Angles: The angles inside the polygon at each vertex<\/p>\n

      Exterior Angles: The angles formed between a side and the extension of an adjacent side<\/p>\n

      2. Convex vs Concave Polygons<\/strong><\/p>\n

      Convex Polygon: All interior angles are less than 180°. The polygon bulges outward. Any line segment connecting two points inside the polygon lies entirely inside the polygon.<\/p>\n

      Example: Square, rectangle, regular pentagon<\/p>\n

      Concave Polygon: At least one interior angle is greater than 180° (reflex angle). The polygon has a "dent" or "cave" inward.<\/p>\n

      Example: Arrowhead shape, a star-shaped polygon<\/p>\n

      Quick Test: If you can draw a line segment between two points inside the polygon that goes outside the polygon, it is concave.<\/p>\n

      3. Regular vs Irregular Polygons<\/strong><\/p>\n

      Regular Polygon: All sides are equal in length, and all interior angles are equal.<\/p>\n

      Examples: Equilateral triangle, square, regular pentagon, regular hexagon<\/p>\n

      Irregular Polygon:<\/strong> Sides are not all equal, or angles are not all equal (or both).<\/p>\n

      Examples: Rectangle (sides not all equal), scalene triangle, any shape that is not regular<\/p>\n

      4. Sum of Interior Angles of a Polygon<\/strong><\/p>\n

      The sum of the interior angles of a polygon depends only on the number of sides (n).<\/p>\n

      Formula: Sum of interior angles = (n – 2) × 180°<\/p>\n

      Why this works: A polygon with n sides can be divided into (n – 2) triangles, and each triangle has an angle sum of 180°.<\/p>\n

      Example 1 – Triangle (n=3): (3 – 2) × 180° = 1 × 180° = 180°<\/p>\n

      Example 2 – Quadrilateral (n=4): (4 – 2) × 180° = 2 × 180° = 360°<\/p>\n

      Example 3 – Pentagon (n=5): (5 – 2) × 180° = 3 × 180° = 540°<\/p>\n

      Example 4 – Hexagon (n=6): (6 – 2) × 180° = 4 × 180° = 720°<\/p>\n

      5. Measure of Each Interior Angle of a Regular Polygon<\/strong><\/p>\n

      Since all interior angles are equal in a regular polygon:<\/p>\n

      Formula: Each interior angle = [(n – 2) × 180°] \/ n<\/p>\n

      Example 1 – Regular pentagon (n=5): (5-2)×180°\/5 = 3×180°\/5 = 540°\/5 = 108°<\/p>\n

      Example 2 – Regular hexagon (n=6): (6-2)×180°\/6 = 4×180°\/6 = 720°\/6 = 120°<\/p>\n

      Example 3 – Regular octagon (n=8): (8-2)×180°\/8 = 6×180°\/8 = 1080°\/8 = 135°<\/p>\n

      6. Sum of Exterior Angles of a Polygon<\/strong><\/p>\n

      The sum of the exterior angles of any polygon (convex) is always 360°, regardless of the number of sides.<\/p>\n

      Important: Take one exterior angle at each vertex (the angle formed by extending one side). The sum is always 360°.<\/p>\n

      Example: For a pentagon, if you extend each side and measure the exterior angles, they add up to 360°.<\/p>\n

      7. Each Exterior Angle of a Regular Polygon<\/strong><\/p>\n

      Since all exterior angles are equal in a regular polygon:<\/p>\n

      Formula: Each exterior angle = 360° \/ n<\/p>\n

      Relationship: Each interior angle + each exterior angle = 180° (they are supplementary)<\/p>\n

      Example 1 – Regular pentagon: Each exterior angle = 360°\/5 = 72° (check: interior 108° + exterior 72° = 180°)<\/p>\n

      Example 2 – Regular hexagon:<\/strong> Each exterior angle = 360°\/6 = 60° (interior 120° + 60° = 180°)<\/p>\n

      8. Number of Diagonals in a Polygon<\/strong><\/p>\n

      A diagonal connects two non-adjacent vertices.<\/p>\n

      Formula: Number of diagonals = n(n – 3)\/2<\/p>\n

      Example 1 – Quadrilateral (n=4): 4(4-3)\/2 = 4×1\/2 = 2 diagonals<\/p>\n

      Example 2 – Pentagon (n=5): 5(5-3)\/2 = 5×2\/2 = 5 diagonals<\/p>\n

      Example 3 – Hexagon (n=6): 6(6-3)\/2 = 6×3\/2 = 9 diagonals<\/p>\n

      Solved Examples<\/strong><\/p>\n

      Example 1 – Sum of Interior Angles:<\/strong> Find the sum of interior angles of an octagon.<\/p>\n

      Solution:<\/strong> n = 8, Sum = (8 – 2) × 180° = 6 × 180° = 1080°<\/p>\n

      Answer:<\/strong> 1080°<\/p>\n

       <\/p>\n

      Example 2 – Each Interior Angle:<\/strong> Find each interior angle of a regular decagon (10 sides).<\/p>\n

      Solution:<\/strong> Sum = (10 – 2) × 180° = 8 × 180° = 1440°; Each = 1440°\/10 = 144°<\/p>\n

      Answer:<\/strong> 144°<\/p>\n

       <\/p>\n

      Example 3 – Each Exterior Angle:<\/strong> Find each exterior angle of a regular hexagon.<\/p>\n

      Solution:<\/strong> Each exterior angle = 360°\/6 = 60°<\/p>\n

      Answer:<\/strong> 60°<\/p>\n

       <\/p>\n

      Example 4 – Finding Number of Sides:<\/strong> Each interior angle of a regular polygon is 150°. How many sides does it have?<\/p>\n

      Solution:<\/strong> Each interior = [(n-2)×180°]\/n = 150°<\/p>\n

      Multiply both sides by n: (n-2)×180 = 150n<\/p>\n

      180n – 360 = 150n<\/p>\n

      30n = 360 → n = 12<\/p>\n

      Answer:<\/strong> 12 sides<\/p>\n

      Common Mistakes to Avoid<\/u><\/strong><\/p>\n

      Mistake 1 – Thinking a circle is a polygon<\/strong>
      \nA circle has a curved boundary, not straight line segments.
      \nCorrect understanding: Polygons have only straight sides.<\/p>\n

      Mistake 2 – Using the wrong formula for interior angles<\/strong>
      \nSum of interior angles = (n – 2) × 180°, not (n – 2) × 360° or n × 180°.
      \nCorrect understanding: Memorize the correct formula.<\/p>\n

      Mistake 3 – Confusing convex and concave<\/strong>
      \nAll regular polygons are convex, but not all convex polygons are regular.
      \nCorrect understanding: Convex means all interior angles < 180°; concave means at least one angle > 180°.<\/p>\n

      Mistake 4 – Forgetting that a square is a regular polygon<\/strong>
      \nSome students think only equilateral triangle is regular.
      \nCorrect understanding: Any polygon with all sides equal AND all angles equal is regular.<\/p>\n

      Mistake 5 – Misidentifying a rectangle as regular<\/strong>
      \nA rectangle has all angles 90°, but sides are not all equal (unless it is a square).
      \nCorrect understanding: A square is regular; a rectangle that is not a square is irregular.<\/p>\n

      Mistake 6 – Incorrectly calculating number of diagonals<\/strong>
      \nUsing n(n-3) instead of n(n-3)\/2 counts each diagonal twice.
      \nCorrect understanding: Divide by 2 because each diagonal has two endpoints.<\/p>\n

      Quick Reference Summary<\/u><\/strong><\/p>\n

      Polygon: Closed 2D figure with straight sides (3 or more sides)<\/p>\n

      Convex Polygon: All interior angles < 180° (bulges outward)<\/p>\n

      Concave Polygon: At least one interior angle > 180° (has a dent)<\/p>\n

      Regular Polygon: All sides equal AND all angles equal<\/p>\n

      Sum of Interior Angles: (n – 2) × 180°<\/p>\n

      Each Interior Angle (Regular): [(n – 2) × 180°] \/ n<\/p>\n

      Sum of Exterior Angles: Always 360°<\/p>\n

      Each Exterior Angle (Regular): 360° \/ n<\/p>\n

      Number of Diagonals: n(n – 3)\/2<\/p>\n

      Common Polygons: Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6), Octagon (8)<\/p>\n

       <\/p>\n

      <\/p>\n