Area of Circles
— the area of a circle equals pi times the radius squared.
Introduction
Past Knowledge
Circumference (11.2.1). Regular polygon area (11.1.3).
Today's Goal
Derive and apply A = πr².
Future Success
Sectors (11.2.4), geometric probability (11.2.5).
Key Concepts
Area Formula
Always use the radius, not the diameter. If given d, divide by 2 first.
Derivation
Think of a circle as a regular polygon with infinitely many sides. As :
The apothem approaches the radius, and the perimeter approaches the circumference.
∎ A = ½aP becomes A = ½ · r · 2πr = πr².
Worked Examples
Given Radius
r = 6
A = 36π ≈ 113.1
Given Diameter
d = 20
r = 10.
A = 100π ≈ 314.2
Find r from Area
A = 49π. Find r.
→ →
r = 7
Common Pitfalls
Squaring the Diameter
A = πr², not πd². If given d = 10, use r = 5 → A = 25π, NOT 100π.
Units Are Squared
Area is in square units (cm², in²). Circumference is in linear units (cm, in).
Real-Life Applications
Pizza Pricing
A 16" pizza has 4× the area of an 8" pizza (not 2×). Area scales with radius squared — doubling the diameter quadruples the pizza.
Sprinkler Coverage
A sprinkler with a 20-foot reach covers A = π(20)² = 400π ≈ 1,257 sq ft of lawn.
Practice Quiz
Loading...