Overview
This lecture explains how to find the circumference and area of circles using formulas based on Pi, with step-by-step examples and key differences highlighted.
Circle Formulas
- The formula for circumference is: Circumference = Pi × diameter (C = π × d).
- The formula for area is: Area = Pi × radius squared (A = π × r²).
- Circumference uses the diameter, while area uses the radius squared.
- The diameter is twice the radius (d = 2r).
- Circumference can also be written as C = π × 2 × r.
Squaring vs. Doubling the Radius
- "Squaring" means multiplying a number by itself (e.g., 3² = 3 × 3).
- For area, square the radius and then multiply by Pi: A = π × r × r.
- Doubling the radius (for circumference) is not the same as squaring it.
Units and Dimensions
- Circumference is measured in linear (1-dimensional) units.
- Area is measured in square (2-dimensional) units.
- Squaring the radius results in square units, which matches the 2-D nature of area.
Example Problems
- Given a circle with radius 8 meters:
- Diameter = 2 × 8 = 16 meters.
- Circumference = 16 × 3.14 = 50.24 meters.
- Area = (8 × 8) × 3.14 = 64 × 3.14 = 200.96 m².
- Earth's equator (diameter ≈ 12,750 km):
- Circumference = 12,750 × 3.14159 = 40,055 km.
- Pizza with a diameter of 24 inches:
- Radius = 24 ÷ 2 = 12 inches.
- Area = (12 × 12) × 3.14 = 144 × 3.14 = 452.16 in².
Key Terms & Definitions
- Pi (π) — The ratio of a circle’s circumference to its diameter, approximately 3.14.
- Circumference — The distance around a circle.
- Diameter — A straight line passing through the center of a circle, touching both sides.
- Radius — The distance from the center to any point on the circle; half the diameter.
- Squaring — Multiplying a number by itself.
Action Items / Next Steps
- Memorize the formulas for circumference (C = πd) and area (A = πr²).
- Practice solving circle problems using provided formulas.
- Complete exercise problems to reinforce understanding.