About

Miscalculating a circle's properties propagates errors through every downstream computation - pipe cross-sections, wheel tolerances, irrigation coverage, orbital mechanics. This calculator derives all principal circle metrics from a single known value: r (radius), d (diameter), C (circumference), or A (area). It uses π at full IEEE 754 double-precision (15 significant digits) rather than the truncated 3.14 approximation that introduces measurable drift in engineering contexts. An optional central angle θ unlocks arc, sector, segment, and chord calculations.

The tool assumes a Euclidean plane. Results deviate on curved surfaces (spherical or hyperbolic geometry). For physical applications, account for manufacturing tolerances. Pro tip: when computing pipe flow area, use the inner radius, not the nominal diameter.

Formulas

All circle properties derive from a single parameter: the radius r. The fundamental relationships are:

A = πr²

C = 2πr

d = 2r

When a central angle θ (in degrees) is provided, partial-circle metrics become available:

L = θ360 × 2πr

A_s = θ360 × πr²

A_seg = r²2 ⋅ (θ_rad − sin(θ_rad))

c = 2r ⋅ sin(θ_rad2)

Where r = radius, d = diameter, C = circumference, A = area, L = arc length, A_s = sector area, A_seg = segment area, c = chord length, θ = central angle in degrees, θ_rad = central angle in radians, and π ≈ 3.14159265358979.

Inverse derivations used when the input is not radius: r = d2, r = C2π, r = √Aπ.

Reference Data

Property	Symbol	Formula	Unit (SI)	Notes
Radius	r	Given or derived	m	Primary input; all others derive from this
Diameter	d	d = 2r	m	Longest chord through center
Circumference	C	C = 2πr	m	Perimeter of the circle
Area	A	A = πr²	m²	Enclosed planar region
Arc Length	L	L = θ360 × 2πr	m	Requires central angle θ
Sector Area	A_s	A_s = θ360 × πr²	m²	"Pizza slice" region
Segment Area	A_seg	A_seg = r²2(θ_rad − sin(θ_rad))	m²	Region between chord and arc
Chord Length	c	c = 2r ⋅ sin(θ2)	m	Straight line between arc endpoints
Pi	π	3.14159265358979…	-	Ratio of circumference to diameter
Semicircle Area	A_½	πr²2	m²	θ = 180°
Quadrant Area	A_¼	πr²4	m²	θ = 90°
Inscribed Square Side	a_in	a_in = r√2	m	Largest square fitting inside the circle
Circumscribed Square Side	a_out	a_out = 2r	m	Smallest square containing the circle
Annulus Area	A_ann	π(R² − r²)	m²	Requires outer radius R
Radian Conversion	θ_rad	θ_rad = θ ⋅ π180	rad	Degrees to radians
Full Revolution	-	360° = 2π rad	-	Complete rotation

Frequently Asked Questions

JavaScript uses IEEE 754 double-precision floats, providing approximately 15-17 significant decimal digits. For radii beyond 10¹⁵ or below 10^-15, rounding errors accumulate - particularly in the area formula where squaring amplifies relative error. This calculator displays up to 10 decimal places. For sub-nanometer or astronomical scales, consider arbitrary-precision libraries.

The segment area formula A_seg = (r² ÷ 2)(θ_rad − sin(θ_rad)) approaches zero as θ approaches 0° because the Taylor expansion of sin(x) ≈ x − x³/6 makes the difference vanishingly small. The result is always non-negative for 0 ≤ θ ≤ 360°. Negative values indicate an input error.

No. All formulas assume a flat Euclidean plane. A circle on a sphere (spherical cap) has area 2πR_sphereh, which differs from πr². For an ellipse, use A = πab where a and b are semi-axes. This tool does not handle those cases.

A sector is the "pizza slice" bounded by two radii and the arc between them. A segment is the region between a chord and the arc it subtends. The segment area equals the sector area minus the triangle formed by the two radii and the chord: A_seg = A_s − 12r²sin(θ_rad).

The conversion is θ_rad = θ_deg × π ÷ 180. Common values: 90° = π/2 rad, 180° = π rad, 360° = 2π rad. This calculator accepts degrees and converts internally.

Yes. For θ > 180°, the chord length represents the straight-line distance across the major arc's endpoints. The chord reaches its maximum (c = 2r, the diameter) at θ = 180°, then decreases symmetrically back to 0 at 360°.