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} ).

Test your knowledge of area formulas for various geometric shapes including triangles, rectangles, and circles. Additionally, explore the different types of polygons and their properties, such as convex, concave, and regular forms.