Area of Shaded Regions & Rings Practice Problems & Conceptual Mastery
To find the area of a shaded region when a shape has an interior hole or cutout removed from it, use the Subtraction Principle: Area of Shaded Region = Area of Total Outer Figure − Area of Inner Unshaded Cutout.
📝 Worked Step-by-Step Example
1. Outer area: 3.14 × 7² = 3.14 × 49 = 153.86. 2. Inner area: 3.14 × 3² = 3.14 × 9 = 28.26. 3. Subtract: 153.86 − 28.26 = 125.6 cm².
🧠 Core Problem-Solving Strategies
1. The Universal Subtraction Formula
A_shaded = A_outer − A_inner. Calculate the area of the entire outer container first, then subtract the unshaded cutout.
2. The Annulus Formula for Rings
For concentric circular rings: Area = πR² − πr² = π(R² − r²). Square both radii first, subtract them, then multiply by π.
3. Never Subtract Radii Directly
Warning: π(R − r)² is NOT equal to π(R² − r²)! You must square each radius individually before subtracting.
📖 Key Mathematical Terms & Vocabulary
Annulus
Concentric Circles
❓ Frequently Asked Questions
What is an annulus in geometry?
An annulus is the Latin word for 'little ring.' In geometry, it is the 2D area between two circles that share the same center point.
Why can't I just calculate π × (R − r)²?
Because (R − r)² = R² − 2Rr + r², which is much smaller than R² − r²! For R = 5 and r = 3: 5² − 3² = 25 − 9 = 16, whereas (5 − 3)² = 2² = 4.
Can this method work for a square with a circular hole?
Yes! Calculate the square area (s²), calculate the circle area (πr²), and subtract: s² − πr².
Continue Learning & Practicing Area of Shaded Regions & Rings
Reinforce your understanding with printable worksheets, step-by-step guides, calculators, and speed games: