Basic Geometry Concepts and Formulas

Overview

• Lecture covers basic geometry: angles, triangles, circles, pi, circumference, polygons, perimeters, lines, plane shapes, and solid shapes.
• Definitions, classifications, key properties, and calculation formulas provided with practical examples.

Angles

• Definition: Space between two straight lines (segments) that meet at a point (vertex).
• Measured in degrees; indicates how open an angle is.
• Classifications:
  • Acute: < 90°
  • Right: = 90°
  • Obtuse: > 90° and < 180°
  • Flat: = 180°
  • Reflex: > 180° and < 360°
  • Full Rotation: = 360°

Triangles

• Definition: Polygon with three sides and three vertices; sum of interior angles = 180°.
• Classification by sides:
  • Equilateral: three equal sides
  • Isosceles: two equal sides
  • Scalene: all sides different
• Classification by angles:
  • Acute triangle: three acute angles
  • Right triangle: one right angle
  • Obtuse triangle: one obtuse angle

Circle and Circumference

• Circumference: curved closed line; all points equidistant from center.
• Circle: plane region bounded by circumference (area inside).
• Difference: circumference = boundary line; circle = interior + boundary.
• Elements:
  • Center: fixed point equidistant to circumference points
  • Radius: segment center to a point on circumference
  • Diameter: segment through center connecting two circumference points; diameter = 2 × radius
  • Chord: segment connecting two points on circumference
  • Arc: portion of circumference between two points
  • Sector: region between two radii and their arc

Number Pi (π)

• Definition: Constant ratio of circumference (perimeter) to diameter of any circle.
• Approximate value: π ≈ 3.14 (commonly used); exact value is infinite non-repeating decimal.
• Test: circumference ÷ diameter ≈ 3.14 for any circle.
• Practical use: essential in architecture, mechanics, engineering.

Circumference Length Formula

• Formulas:
  • Circumference = 2 × π × radius
  • Circumference = π × diameter
• Use π = 3.14 for examples in lecture.
• Example calculations:
  • Radius 2.36 in → Circumference = 2 × 3.14 × 2.36 = 14.82 in
  • Box radius 4 in → String needed = 2 × 3.14 × 4 = 25.12 in
  • Fountain diameter 29.5 ft → Fence = 3.14 × 29.5 = 92.71 ft

Polygons

• Definition: Plane figure described by a closed polygonal line (straight segments).
• Parts:
  • Sides: line segments forming polygon
  • Vertices: intersection points of sides
  • Angles: between two adjacent sides
  • Diagonals: segments connecting non-adjacent vertices
• Classification:
  • Regular: all sides and angles equal
  • Irregular: sides and/or angles not equal
• Classification by number of sides (examples):
  • Triangle: 3 sides
  • Quadrilateral: 4 sides
  • Pentagon: 5 sides
  • Hexagon: 6 sides
  • Heptagon: 7 sides
  • Octagon: 8 sides

Perimeter Of Polygons

• Definition: Sum of the lengths of all sides; measured in linear units.
• Strategy:
  • Add all side lengths
  • For regular shapes, multiply side length by number of sides
• Examples:
  • Triangle sides 12, 14, 10 → Perimeter = 12 + 14 + 10 = 36 in
  • Square side 20 ft → Perimeter = 4 × 20 = 80 ft
  • Rectangle 6 ft by 10 ft → Perimeter = 2×6 + 2×10 = 32 ft
  • Irregular pool with sides 16, 36, 29, 23, 13, 13 → Perimeter = 130 ft

Types Of Lines

• Straight lines: horizontal, vertical, oblique.
  • Horizontal: parallel to horizon.
  • Vertical: up-down direction.
  • Oblique: slanted.
• Special relations:
  • Parallel lines: never meet; maintain equal distance.
  • Perpendicular lines: intersect at right angles.
• Polygonal lines: straight lines joined; can be open or closed.
• Curved lines: can be open (e.g., spiral) or closed; include wavy and spiral curves.

Plane Shapes (Basic)

• Circle: round; boundaries are circumference.
• Oval: elongated circle, e.g., egg or rugby ball.
• Triangle: 3 sides, 3 angles.
• Square: 4 equal sides, 4 equal angles.
• Rectangle: 4 sides; opposite sides equal.
• Diamond (rhombus): 4 equal sides, unequal angles.
• Pentagon: 5 sides.
• Hexagon: 6 sides.

Solid (3D) Shapes

• Sphere: all points equidistant from center (tennis ball).
• Cube: six square faces; all edges equal (dice, box).
• Cylinder: two identical circular bases (can, candle).
• Prism: two identical polygonal faces (milk carton as rectangular prism).
• Pyramid: polygonal base, triangular faces meet at apex (Egyptian pyramid).
• Cone: circular base and a vertex (ice cream cone, party hat).

Key Terms And Definitions

• Vertex: point where two segments meet.
• Segment: part of a straight line between two endpoints.
• Radius: center to circumference segment.
• Diameter: segment through center connecting opposite circumference points.
• Chord: segment between two circumference points.
• Arc: curved portion of circumference.
• Sector: wedge-shaped region inside circle.
• Perimeter: total length around a figure.

Action Items / Next Steps

• Practice: identify shapes and elements around home; measure circumferences and verify π ≈ 3.14.
• Exercises:
  • Calculate circumferences using 2πr and πd with given examples.
  • Find perimeters of rooms or objects using measuring tape.
  • Classify polygons by sides and regularity using everyday objects.