Area of a Circle
\text{width} = \frac{1}{2} \times 2\pi r = \pi rTranscript
Today we discover why a circle's area is pi r squared. Imagine slicing a circle into many thin pizza slices. Let me draw a circle and cut it into eight wedges.
- radius = r
If we rearrange these wedges, they almost form a rectangle. The width of this rectangle is half the circle's perimeter.
- Wedges → rectangle shape
\text{width} = \frac{1}{2} \times 2\pi r = \pi rThe height of the rectangle is the radius r.
\text{height} = rSo the area is width times height: pi r times r. That gives us area equals pi r squared.
A = \pi r^2Let's try a real example: a circle with radius three. Area equals pi times three squared, which is nine pi.
- Example: r = 3
A = \pi \times 3^2 = 9\pi \approx 28.3That's about twenty-eight point three square units. Quick check: if radius doubles, does area double or quadruple? Remember: area of a circle is always pi r squared.
- 9π ≈ 28.3 sq units
Quick quiz
Q1. If radius doubles from r to 2r, area changes by what factor?
Area goes from πr² to π(2r)² = 4πr², so it quadruples.
