Polygons and Quadrilaterals

Polygons

• A polygon is a 2D shape with straight sides that are fully closed.
• Sides are called edges, and points where two edges meet are vertices.
• Polygons are named by the number of sides:
  • Triangle (3 sides)
  • Quadrilateral (4 sides)
  • Pentagon (5 sides)
  • Hexagon (6 sides)
  • Heptagon (7 sides)
  • Octagon (8 sides)
  • Nonagon (9 sides)
  • Decagon (10 sides)
• Regular polygons have equal sides and angles.
• Irregular polygons have unequal sides and/or angles.

Quadrilaterals

• A quadrilateral is a polygon with 4 sides.
• Types of quadrilaterals include:
  • Square: All sides equal, all angles 90°, opposite sides parallel.
  • Rectangle: Opposite sides equal, all angles 90°, opposite sides parallel.
  • Rhombus: All sides equal, opposite sides parallel, opposite angles equal, angles not necessarily 90°.
  • Parallelogram: Opposite sides equal and parallel, opposite angles equal, angles not necessarily 90°.
  • Trapezoid (US)/Trapezium (UK): One pair of opposite sides parallel, other pair not necessarily parallel or equal.
  • Kite: Two pairs of adjacent sides equal, one pair of opposite angles equal, diagonals intersect at a right angle.
• The sum of interior angles in any quadrilateral is 360°.
• Diagonals may or may not be equal, depending on the type.

Solving Problems with Polygons and Quadrilaterals

• Finding the sum of interior angles of a polygon:

  • Formula: (n-2) × 180° (where n is the number of sides).

• Finding each interior angle in a regular polygon:

  • Formula: [(n-2) × 180°] / n

• Finding exterior angles of a polygon:

  • The sum of exterior angles of any polygon is always 360°.

• Finding the exterior angle of a regular polygon: -Formula: 360°/n

• Finding unknown angles in a quadrilateral:

  • The sum of all interior angles in a quadrilateral is 360°. Find the missing angle by subtracting the known angles from 360°.

• Finding perimeter and area of specific quadrilaterals

• Square: Perimeter = 4 × side length; Area = side length².

• Rectangle: Perimeter = 2 × (length + width); Area = length × width.

• Parallelogram: Perimeter = 2 × (base + side); Area = base × height.

• Rhombus: Perimeter = 4 × side length; Area = (diagonal1 × diagonal2)/2.

• Trapezoid: Perimeter = sum of all sides; Area = (1/2) × (base1 + base2) × height

• Properties of diagonals in quadrilaterals:

  • Square: Diagonals equal, bisect at right angles.
  • Rectangle: Diagonals equal, do not bisect at right angles.
  • Rhombus: Diagonals bisect at right angles, not equal.
  • Parallelogram: Diagonals bisect each other, not necessarily equal.
  • Kite: One diagonal bisects the other at a right angle.