AIT 7 Mathematics Lecture on Perimeter, Circumference, Area, and Volume 📐

Introduction

• Focus on perimeter, circumference, area, and volume.
• Concepts explained from 1D to 3D.

Basic Concepts

One-Dimensional (1D)

• Length: Quantifies a 1D object (e.g., 1 cm line).

Two-Dimensional (2D)

• Area: Quantifies a 2D object.
  • Square Example: Extending a 1 cm line perpendicularly creates a 1 cm square (1 square cm).

Three-Dimensional (3D)

• Volume: Quantifies a 3D object.
  • Cube Example: Extending a 1 cm^2 square up by 1 cm creates a 1 cubic cm cube.
• Units: Square units for area (cm^2), cubic units for volume (cm^3).

Perimeter

• Definition: Total distance or length around a shape.
• Units: 1D measure (e.g., cm, inches, miles).
• Calculation: Sum the lengths of all sides.
  • Example: Pentagon with each side 10 m, perimeter = 50 m.
  • Example: Right triangle with sides 3 cm, 4 cm, 5 cm, perimeter = 12 cm.
  • Regular Polygon: Hexagon with each side 4 cm, perimeter = 24 cm (6 * 4).
  • Irregular Polygon: Calculate missing side lengths and sum all sides.

Circles: Circumference and Area

Circumference (C)

• Formula: C = π * D (π = 3.14)
  • Example: Radius 8 cm, diameter = 16 cm, C = 3.14 * 16 = 50.24 cm.

Area (A)

• Formula: A = π * r^2
  • Example: Radius 8 cm, A = 3.14 * 8^2 = 200.96 cm^2.
• Radius vs Diameter: Radius is half the diameter, diameter is twice the radius.

Area of Polygons

Squares and Rectangles

• Formula: Area = length * width.
  • Square Example: 2 cm x 2 cm, area = 4 cm^2.
  • Rectangle Example: 4 cm x 2 cm, area = 8 cm^2.

Triangles

• Formula: Area = 1/2 * base * height.
  • Example: Right triangle with base 3 cm, height 4 cm, area = 6 cm^2.
  • Adjustments: Base and height for different triangles (acute, obtuse).

Parallelograms

• Formula: Area = base * height.
  • Example: Base 5 cm, height 2 cm, area = 10 cm^2.

Trapezoids

• Formula: Area = ((base1 + base2) * height) / 2.
  • Example: Bases 8 cm, 4 cm, height 6 cm, area = 36 cm^2.

Volume of 3D Shapes

Prisms

• Formula: Volume = area of base * height.
  • Square/Rectangular Prism: 4 cm x 3 cm x 10 cm, volume = 120 cm^3.
  • Triangular Prism: Base 10 cm, height 8 cm, length 50 cm, volume = 2000 cm^3.

Cylinders

• Formula: Volume = π * r^2 * height.
  • Example: Radius 2 cm, height 10 cm, volume = 125.6 cm^3.

Cones

• Relationship: Volume is 1/3 that of a cylinder of the same radius and height.
• Formula: Volume = (π * r^2 * height) / 3.
  • Example: Radius 5 cm, height 12 cm, volume = 314 cm^3.

Rectangular Pyramids

• Similar to cones in that volume is 1/3 that of a rectangular prism.
• Formula: Volume = (length * width * height) / 3.
  • Example: Length 8 cm, width 7 cm, height 11 cm, volume = 205.33 cm^3.

Spheres

• Formula: Volume = 4/3 * π * r^3.
  • Example: Radius 10 cm, volume = 4186 cm^3.

Conclusion

• Reviewed key mathematical concepts for the TEAS exams.
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