Circle Calculator — Area, Circumference, Diameter, Arc Length & Sector
Calculate circle area, circumference, diameter, and radius from any known value. Includes sector area, arc length formulas, and real-world applications with worked examples.
Enter any one measurement — radius, diameter, area, or circumference — and get all the others instantly. Circles come up everywhere from wheel design to pizza sizing to orbital mechanics. Use the CalcHub Circle Calculator to solve any circle problem in seconds.
The Core Formulas
| Measurement | Formula |
|---|---|
| Area | A = πr² |
| Circumference | C = 2πr = πd |
| Diameter | d = 2r |
| Radius from area | r = √(A/π) |
| Radius from circumference | r = C/(2π) |
Worked Examples
Given radius = 7 cm:- Area = π × 7² = 49π ≈ 153.94 cm²
- Circumference = 2π × 7 = 14π ≈ 43.98 cm
- Diameter = 14 cm
- Radius = 62.83 / (2π) ≈ 10 m
- Area = π × 10² ≈ 314.16 m²
- r = √(200/π) = √63.66 ≈ 7.98 cm
- Circumference = 2π × 7.98 ≈ 50.12 cm
Arc Length and Sector Area
A sector is the "pie slice" portion of a circle.
Arc Length: L = rθ (θ in radians) or L = (θ/360°) × 2πr (θ in degrees) Sector Area: A = (1/2)r²θ (θ in radians) or A = (θ/360°) × πr² Example: A sector with r = 10 cm and θ = 60°Arc length = (60/360) × 2π × 10 = (1/6) × 62.83 ≈ 10.47 cm
Sector area = (60/360) × π × 10² = (1/6) × 314.16 ≈ 52.36 cm²
Annulus (Ring Between Two Circles)
When a smaller circle is cut from a larger one:
Area of annulus = π(R² − r²) = π(R+r)(R−r)
Where R = outer radius, r = inner radius.
Example: Outer radius 10 cm, inner radius 6 cm. Area = π(100 − 36) = 64π ≈ 201.06 cm²Real-World Applications
| Application | What You Calculate |
|---|---|
| Circular garden fencing | Circumference (length of fence needed) |
| Pizza: 12" vs 16" | Area (how much food you're getting) |
| Wheel travel per rotation | Circumference |
| Sprinkler coverage | Area |
| Pipe internal volume | Area of cross-section × length |
| Clock minute hand travel | Arc length per hour |
One 16" pizza: radius = 8", area = π × 8² = 64π ≈ 201.1 sq in
Two 8" pizzas: radius = 4" each, area = 2 × π × 16 = 32π ≈ 100.5 sq in
The single large pizza has twice the area of two small ones — so if both cost the same, the 16" is dramatically better value.
Unit Conversion Reference
| Shape | Imperial | Metric |
|---|---|---|
| Area of 1 ft radius circle | 3.14 ft² | 0.291 m² |
| Circumference of 1 m circle | 3.28 ft | 6.28 m |
What's the most efficient shape?
The circle has the largest area for a given perimeter of any shape — this is the isoperimetric inequality. That's why soap bubbles are spherical, why eyes are circular, and why many storage tanks are cylindrical: maximum volume, minimum material.
How do I find where two circles intersect?
Set the two circle equations equal and solve. For circles at (h₁,k₁) with radius r₁ and (h₂,k₂) with radius r₂: they intersect if |r₁−r₂| < distance between centers < r₁+r₂. The actual intersection points require solving a system of equations.
Why is the area formula πr² and not something simpler?
Calculus derivation: integrate 2πx from 0 to r (summing up thin rings) and you get πr². The geometric proof goes back to Archimedes, who showed the area equals that of a right triangle with legs r and 2πr: area = (1/2) × r × 2πr = πr².