Geometry Formulas and Polygon Types

Formulas For Area

• Triangle:
  • Area = (1/2) × base × height
• Rectangle:
  • Area = length × width
• Parallelogram:
  • Area = base × height
• Trapezoid:
  • Area = (1/2) × (base1 + base2) × height
• Circle:
  • Area = π × radius²
• Regular Polygon:
  • Area = (1/2) × Perimeter × Apothem
• Pentagon:
  • Area = (1/4) √(5(5+2√5)) × side²
• Hexagon:
  • Area = (3√3/2) × side²

Types Of Polygons

• Convex Polygons:
  • All interior angles < 180°
• Concave Polygons:
  • At least one interior angle > 180°
• Regular Polygons:
  • All sides and angles are equal
• Irregular Polygons:
  • Sides and/or angles are not equal
• Quadrilaterals:
  • Four sides; includes rectangles, squares, trapezoids, etc.
• Triangles:
  • Three sides; categorized as scalene, isosceles, or equilateral
• Pentagons:
  • Five sides; can be regular or irregular
• Hexagons:
  • Six sides; common in nature (e.g., honeycomb)

Properties Of Regular Polygons

• Equal Sides and Angles:
  • All sides and angles are congruent
• Symmetry:
  • Regular polygons are symmetric about their center
• Interior Angle Formula:
  • Interior Angle = (n-2) × 180° / n (where n = number of sides)
• Exterior Angle Formula:
  • Exterior Angle = 360° / n
• Sum of Interior Angles:
  • Sum = (n-2) × 180° (where n = number of sides)
• Apothem:
  • The distance from the center to the midpoint of a side; used in area calculation
• Diagonals:
  • The number of diagonals = n(n-3)/2 (for an n-sided polygon)

Formulas For Area

• Area of a Triangle: Calculated using the formula ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ).
• Rectangle area: Determined by multiplying length and width ( \text{Area} = \text{length} \times \text{width} ).
• For a Parallelogram, area is given by ( \text{Area} = \text{base} \times \text{height} ).
• To find the area of a Trapezoid: Use the formula ( \text{Area} = \frac{1}{2} \times (\text{base1} + \text{base2}) \times \text{height} ).
• A Circle's area is calculated as ( \text{Area} = \pi \times \text{radius}^2 ).
• Area of a Regular Polygon: Use ( \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} ).
• For a Pentagon, the area can be found using ( \text{Area} = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} \times \text{side}^2 ).
• A Hexagon has an area calculated by ( \text{Area} = \frac{3\sqrt{3}}{2} \times \text{side}^2 ).

Types Of Polygons

• Convex Polygons: All interior angles are less than 180°, resulting in no indentations.
• Concave Polygons: At least one interior angle exceeds 180°, creating a "caved-in" effect.
• Regular Polygons: Characterized by equal sides and angles, exhibiting uniformity in shape.
• Irregular Polygons: Have sides and/or angles that vary, lacking uniformity.
• Quadrilaterals: Four-sided polygons, encompassing shapes like rectangles, squares, and trapezoids.
• Triangles: Comprised of three sides, further classified into scalene (unequal sides), isosceles (two equal sides), and equilateral (all sides equal).
• Pentagons: Contain five sides and can be either regular or irregular.
• Hexagons: Feature six sides, often observed in natural forms such as a honeycomb structure.

Properties Of Regular Polygons

• Equal Sides and Angles: Regular polygons maintain congruency among all sides and angles.
• Symmetry: These polygons exhibit symmetry about their center point.
• Interior Angle Formula: ( \text{Interior Angle} = \frac{(n-2) \times 180°}{n} ), where n is the number of sides.
• Exterior Angle Formula: ( \text{Exterior Angle} = \frac{360°}{n} ) helps find the angle outside a polygon.
• Sum of Interior Angles: Given by ( \text{Sum} = (n-2) \times 180° ).
• Apothem: The distance from the center to the midpoint of a side, critical for calculating area in regular polygons.
• Diagonals: The formula for determining the number of diagonals in an n-sided polygon is ( \text{Diagonals} = \frac{n(n-3)}{2} ).