Circle Area, Perimeter, and Sector Calculator 🌎

Circle Area Formula

The area enclosed by a circle of radius r:

Area = π × r²

Where π ≈ 3.14159265358979. This formula gives area in square units (cm², m², in², etc.).

Perimeter (Circumference)

The total distance around the circle:

Perimeter = 2 × π × r = π × d

Where d = 2r is the diameter. Knowing the diameter? Just multiply it by π.

Circular Sector — Area and Arc

A sector is the "pie slice" region between two radii that form angle θ (in degrees):

Sector Area  = (θ / 360) × π × r²
Arc Length   = (θ / 360) × 2 × π × r

For θ in radians: Sector Area = ½ × r² × θ and Arc Length = r × θ.

Reference Table: Common Radii

Key insight: doubling the radius quadruples the area (area scales with r²), but only doubles the perimeter.

Sector Reference Table (radius = 10)

Finding Radius from Area or Perimeter

If you know the area: r = √(Area / π)
If you know the perimeter: r = Perimeter / (2π)

Example: area = 200 → r = √(200 / 3.14159) = √63.66 ≈ 7.98.

Why π?

π (pi) is the ratio of any circle's circumference to its diameter — constant for every circle, regardless of size. It's irrational (infinite non-repeating decimals) and transcendental (not a root of any polynomial with integer coefficients).

Practical precision: π ≈ 3.1416 is sufficient for most engineering. For software or science, use the full double-precision value (15–17 significant digits).

Degrees vs. Radians

360° = 2π radians
1°   = π/180 ≈ 0.01745 rad
1 rad = 180°/π ≈ 57.296°

This calculator accepts degrees because they are more intuitive. The formula automatically converts internally.