A = πr². You've used it a hundred times. But do you know why it works? And more importantly, do you know the mistakes that cost points every time?
The Formula
A = πr²
Where:
- A = area of the circle
- π ≈ 3.14159
- r = radius (half the diameter)
Critical: The radius must be SQUARED. Not multiplied by 2. Not multiplied by π. Squared first, then multiplied by π.
Why Does πr² Work? The Visual Proof
Imagine a circle as a pizza. Cut it into many thin slices (say, 16). Rearrange them alternately up-down:
The slices form something that looks like a rectangle:
- Height = radius (r)
- Width = half the circumference = πr
- Area of rectangle = r × πr = πr²
More slices → more rectangular → exact answer in the limit.
Step-by-Step Calculations
Example 1: r = 4
A = π(4²) = π(16) = 50.27 sq units
Example 2: d = 14 (must find r first!)
r = 14/2 = 7
A = π(7²) = π(49) = 153.94 sq units
Example 3: C = 44 (must find r first!)
r = 44/(2π) = 44/6.283 = 7.003 ≈ 7
A = π(49) = 153.94 sq units
Example 4: A = 314.16 given, find r
r = √(314.16/π) = √(100) = 10
The Three Fatal Mistakes
Mistake 1: Using Diameter Instead of Radius
Wrong: A = π(10²) = 314.16 (when d = 10)
Right: r = 5, A = π(5²) = 78.54
The wrong answer is 4× too large because (2r)² = 4r².
Mistake 2: Forgetting to Square
Wrong: A = π(5) = 15.71
Right: A = π(25) = 78.54
The wrong answer is 5× too small.
Mistake 3: Using 2πr (Circumference Formula)
Wrong: A = 2π(5) = 31.42
Right: A = π(25) = 78.54
2πr is circumference, NOT area. Area = πr².
Real-World Examples
Pizza Value Comparison
| Size | Diameter | Radius | Area | Price | Price/sq in | |------|----------|--------|------|-------|------------| | Small 10" | 10 | 5 | 78.5 | ₹200 | ₹2.55 | | Medium 12" | 12 | 6 | 113.1 | ₹300 | ₹2.65 | | Large 14" | 14 | 7 | 153.9 | ₹350 | ₹2.27 |
The large pizza is the best value per square inch.
Circular Garden Bed
Radius 3m garden: A = π(9) = 28.27 sq m
Soil needed (15cm deep): 28.27 × 0.15 = 4.24 cubic meters
Sprinkler Coverage
Sprinkler radius 6m: A = π(36) = 113.1 sq m
To cover a 20m × 30m lawn (600 sq m), you need 600/113.1 ≈ 6 sprinklers with overlap.
The Trench Truth: Area scales with the square of the radius. Double the radius → 4× the area. Triple the radius → 9× the area. This is why a 14" pizza is 36% more pizza than a 12" — not just 17% more. The non-linear scaling surprises most people.
Calculate any circle's area instantly with our circle calculator.
Related: Area Converter · Square Root Calculator · Quadratic Formula Calculator
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