The area of a circle is the amount of two-dimensional space enclosed by the circle's boundary. The formula A = πr² is one of the most fundamental results in geometry, connecting a circle's radius to the flat space it occupies.
Why πr² Works — A Visual Proof
Imagine slicing a circle into many thin sectors (like pizza slices) and rearranging them alternately to form an approximate parallelogram. As the number of slices increases, the shape converges to a rectangle with height r (the radius) and width πr (half the circumference). The area of this rectangle is r × πr = πr². This sector-rearrangement proof was first demonstrated by Archimedes and remains the most intuitive explanation.
The Formula and Its Variables
Where: A = area, π ≈ 3.14159, r = radius (distance from center to edge)
If you know the diameter (d) instead, use r = d/2, giving A = π(d/2)² = πd²/4.
Step-by-Step Calculation
- Identify the radius — measure from the center to any point on the circumference.
- Square it — multiply the radius by itself (r²).
- Multiply by π — use 3.14159 or your calculator's π key.
Worked Examples
Example 1: Simple radius
A circle has radius 5 cm. A = π × 5² = π × 25 = 78.54 cm²
Example 2: Diameter given
Circular garden, diameter 8 m. r = 4 m → A = π × 16 = 50.27 m²
Example 3: Real-world (pizza)
14-inch pizza: r = 7 in → A = π × 49 = 153.94 in²
Common Mistakes
- Using diameter instead of radius — always halve the diameter first.
- Forgetting to square — writing πr instead of πr².
- Wrong units — if r is in cm, area is cm².
Real-World Applications
Pizza: A 16-inch pizza has 4× the area of an 8-inch one. Garden: Calculate mulch needed. Pipes: Doubling radius quadruples flow area.
Related Calculators and Articles
Use the Circle Area Calculator for instant results. See the Sphere Calculator, Circumference Calculator, Area Formulas, and How to Calculate Area.