📖 Learn
1. Area of a Circle
A = πr²
r = radius (half the diameter)
Example: r = 5 cm → A = π × 5² = 25π ≈ 78.54 cm²
From diameter: r = d/2 first, then A = π(d/2)²
Exact answer: leave in terms of π (e.g. 25π cm²)
💡 r² means r × r — don't forget to square the radius BEFORE multiplying by π.
⚠️ Common error: A = π × 2r (wrong!) or A = (2πr)² (wrong!). It's always πr².
2. Finding Radius from Area
Rearrange: A = πr² → r² = A/π → r = √(A/π)
Example: A = 78.54 cm² → r² = 78.54/π ≈ 25 → r = 5 cm
Example: A = 50 cm² → r = √(50/π) ≈ 3.99 cm
💡 Divide by π first, then take the square root.
3. Area of a Sector
A sector is a "pizza slice" — fraction of a circle based on angle θ.
Area of sector = (θ / 360) × πr²
Example: r = 8 cm, θ = 90°: Area = (90/360) × π × 64 = 16π ≈ 50.27 cm²
Semicircle: (180/360) × πr² = ½πr²
💡 For a quarter circle (quadrant), area = ¼πr²
4. Annulus (Ring Shape)
An annulus is the region between two concentric circles (a ring).
Area of annulus = πR² − πr² = π(R² − r²)
R = outer radius, r = inner radius
Example: R = 10, r = 6: Area = π(100−36) = 64π ≈ 201.1 cm²
5. Composite Shapes with Circles
Circle inside a square: Shaded area = square area − circle area
Semicircle on rectangle: Total area = rectangle + ½πr²
Example: 10×8 rectangle + semicircle r=4 on top: Area = 80 + ½π(16) = 80+8π ≈ 105.1 cm²
⚠️ Carefully identify what is being ADDED and what is being SUBTRACTED.