Circle Area and Circumference Calculator
This calculator finds all three fundamental measurements of a circle from just its radius (r). Enter r and get the area (A = π · r²), the circumference (C = 2 · π · r — the total boundary length, also called the perimeter), and the diameter (D = 2r). These formulas are foundational in geometry, engineering, architecture, and everyday tasks such as sizing pipes, circular land plots, wheels, pools, and pizza pans. The constant π ≈ 3.14159265358979 links every circle's dimensions universally.
Circle area = π · r² ≈ 3.14159 · r². Circumference = 2 · π · r ≈ 6.28318 · r. For r = 5 cm: area = 78.54 cm², circumference = 31.42 cm, diameter = 10 cm. Doubling the radius quadruples the area.
When to use this calculator
- Calculating the area of a circular garden bed or irrigation zone to determine how much mulch, sod, or fertilizer to purchase.
- Determining the circumference of a wheel or tire to compute distance traveled per revolution in automotive or bicycle engineering.
- Sizing circular pipes, tanks, or silos in plumbing and civil engineering — cross-sectional area directly affects flow rate (Q = A · v).
- Finding the material needed to cut circular pieces from sheet metal, fabric, glass, or plywood — minimizing waste in manufacturing.
- Computing the footprint of a round table, fountain, or column base for interior design or construction layout purposes.
- Estimating the water volume of a circular swimming pool: volume = area × depth.
Worked Example: Circular Patio with r = 5 m
- Radius r = 5 m
- Area A = π × 5² = 3.14159 × 25 = 78.54 m²
- Circumference C = 2 × π × 5 = 6.28318 × 5 = 31.42 m
- Diameter D = 2 × 5 = 10 m
- At $6/m² concrete cost → patio cost ≈ $471; border edging at $4/m → ≈ $126
How it works
2 min readFormulas
All outputs derive from a single input — the radius r (distance from center to any point on the boundary):
Diameter D = 2 · r
Area A = π · r²
Circumference C = 2 · π · r (also written π · D)
π ≈ 3.14159265358979…These formulas are exact for a perfect Euclidean circle. This calculator uses IEEE 754 double-precision, giving results accurate to ~15 significant digits.
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Quick Reference Table
| Radius (r) | Diameter (D) | Area (A) | Circumference (C) |
|---|---|---|---|
| 1 cm | 2 cm | 3.14 cm² | 6.28 cm |
| 2 cm | 4 cm | 12.57 cm² | 12.57 cm |
| 3 cm | 6 cm | 28.27 cm² | 18.85 cm |
| 5 cm | 10 cm | 78.54 cm² | 31.42 cm |
| 7 cm | 14 cm | 153.94 cm² | 43.98 cm |
| 10 cm | 20 cm | 314.16 cm² | 62.83 cm |
| 15 cm | 30 cm | 706.86 cm² | 94.25 cm |
| 20 cm | 40 cm | 1,256.64 cm² | 125.66 cm |
| 25 cm | 50 cm | 1,963.50 cm² | 157.08 cm |
| 50 cm | 1 m | 7,853.98 cm² | 314.16 cm |
| 1 m | 2 m | 3.14 m² | 6.28 m |
| 3 m | 6 m | 28.27 m² | 18.85 m |
| 5 m | 10 m | 78.54 m² | 31.42 m |
| 10 m | 20 m | 314.16 m² | 62.83 m |
> Key rule: Area scales with r² — doubling the radius quadruples the area. Circumference scales linearly — doubling the radius doubles the circumference.
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Worked Examples
Example 1 — Circular Patio (r = 5 ft)
A homeowner wants a round concrete patio with radius 5 ft.
A = π · 5² = 78.54 ft²
C = 2 · π · 5 = 31.42 ft (border length)
D = 10 ftAt $6/ft² concrete → slab ≈ $471; decorative border at $4/linear ft → ~$126.
Example 2 — Bicycle Wheel (r = 13.5 in)
A standard 27-inch road wheel has r = 13.5 in.
C = 2 · π · 13.5 ≈ 84.82 in ≈ 7.07 ft per revolutionAt 90 RPM, that equals ~6.36 ft/s ≈ 4.3 mph — useful for speedometer calibration.
Example 3 — Circular Swimming Pool (r = 4 m)
A = π · 4² ≈ 50.27 m²
Volume (depth 1.5 m) = 50.27 × 1.5 ≈ 75.4 m³ ≈ 75,400 liters---
Common Mistakes
1. Radius vs. diameter confusion — If you measured across the full circle, divide by 2 first. Using diameter as radius inflates the area by 4×.
2. Using π ≈ 3.14 — Acceptable for rough estimates, but for r = 100 m it introduces a 16 m² error. Use full precision for engineering.
3. Mixing units — If r is in inches, A is in in², not cm². Convert before entering.
4. Applying to ellipses — Ovals have two semi-axes; the circle formulas give wrong results for non-circular shapes.
Frequently asked questions
What is the formula for the area of a circle?
A = π · r², where r is the radius and π ≈ 3.14159265. For r = 7 cm: A = π · 49 ≈ 153.94 cm². Proved by Archimedes (~250 BC) using exhaustion; remains exact for any perfect circle.
What is the formula for the circumference (perimeter) of a circle?
C = 2 · π · r = π · D, where D is the diameter. For r = 5 cm: C = 2 × 3.14159 × 5 ≈ 31.42 cm. 'Circumference' and 'perimeter' are synonymous for circles.
What is the difference between radius, diameter, and circumference?
Radius (r) is the distance from center to edge. Diameter (D = 2r) is the longest straight line through the center. Circumference (C = 2πr = πD) is the total length of the boundary curve — the 'perimeter' of the circle.
How do I find the radius if I only know the area?
Rearrange: r = √(A / π). For A = 200 cm²: r = √(200 / 3.14159) = √63.662 ≈ 7.98 cm. Useful when you know the desired coverage area and need the radius.
How do I find the radius if I only know the circumference?
Rearrange: r = C / (2π). For a pipe with circumference 94.25 mm: r = 94.25 / 6.28318 ≈ 15.0 mm, diameter = 30 mm. Standard technique to measure round stock without calipers.
Why does doubling the radius quadruple the area?
Because area depends on r². If r doubles: A = π · (2r)² = 4 · π · r² = 4× the original. A 12-inch pizza has 4× the area of a 6-inch pizza — always think in squared terms when scaling circles.
Can I use this calculator for real-world problems like land plots or pools?
Yes — for any true circle. Circular pool r = 4 m → A ≈ 50.27 m²; at depth 1.5 m, volume ≈ 75,400 liters. Circular land plot r ≈ 117.75 ft covers exactly 1 acre (43,560 ft²). The same π formulas apply everywhere.
What is the relationship between area and circumference?
Both are functions of r: A = C² / (4π) and C = 2√(π · A). If C = 62.83 cm: A = (62.83)² / (4π) ≈ 314.16 cm² (r = 10 cm). Useful when you can only measure one quantity.
What value of π is used in this calculator?
IEEE 754 double-precision, which gives π ≈ 3.141592653589793 (15–17 significant digits). The limiting factor in any practical calculation is the precision of the radius you enter, not the π approximation.