About

Miscalculating a circle's dimensions propagates errors through every dependent measurement - pipe cross-sections, wheel clearances, irrigation coverage, antenna gain patterns. This calculator derives all principal circle metrics from a single known quantity: r (radius), d (diameter), C (circumference), or A (area). It uses IEEE 754 double-precision π ≈ 3.141592653589793, yielding roughly 15 significant digits before floating-point truncation. Results include arc length and sector/segment areas for any central angle θ, which matter in cam design, pizza portioning, and land surveying alike.

The tool assumes a perfect Euclidean circle on a flat plane. For geodesic arcs on Earth's surface, curvature corrections apply at radii above ~10 km. All angular inputs accept degrees; internal computation converts to radians via θ_rad = θ_deg ⋅ π ÷ 180. Pro tip: when measuring physical circles, measure diameter rather than radius - it passes through the center, reducing parallax error by half.

Formulas

All circle measurements derive from a single parameter: the radius r. Given any one of the four base quantities, the calculator inverts to find r, then computes the rest.

d = 2r

C = 2πr

A = πr²

L = rθ

A_s = 12r²θ

c = 2r sin(θ2)

A_seg = 12r²(θ − sinθ)

Inverse relations for recovering r from other knowns:

r = d2

r = C2π

r = √Aπ

Where r = radius, d = diameter, C = circumference, A = area, θ = central angle in radians, L = arc length, A_s = sector area, c = chord length, A_seg = segment area. All angular inputs are internally converted: θ_rad = θ_deg × π ÷ 180.

Reference Data

Property	Symbol	Formula from r	Unit (SI)	Example (r = 5)
Radius	r	-	m	5.000
Diameter	d	2r	m	10.000
Circumference	C	2πr	m	31.416
Area	A	πr²	m²	78.540
Arc Length	L	rθ	m	7.854 (90°)
Sector Area	A_s	12r²θ	m²	19.635 (90°)
Chord Length	c	2r sin(θ÷2)	m	7.071 (90°)
Segment Area	A_seg	12r²(θ − sinθ)	m²	7.135 (90°)
Sagitta (Height)	h	r(1 − cos(θ÷2))	m	1.464 (90°)
Inscribed Square Side	a_in	r√2	m	7.071
Circumscribed Square Side	a_out	2r	m	10.000
Inscribed Hex Side	s_hex	r	m	5.000
Curvature	κ	1r	m⁻¹	0.200
Sphere Surface (same r)	S	4πr²	m²	314.159
Sphere Volume (same r)	V	43πr³	m³	523.599
π Approximation	π	-	-	3.14159265358979

Frequently Asked Questions

The tool uses IEEE 754 double-precision floating point, giving roughly 15-16 significant digits for π. Textbooks typically truncate to 3-5 digits. You can use the precision selector to match your required significant figures. For engineering work, 4-6 digits usually suffice; for CNC machining tolerances below 0.001 mm, use maximum precision.

Sector area A_s = 12r²θ includes the triangular portion from center to chord. Segment area subtracts that triangle: A_seg = A_s − 12r²sinθ. At 180°, the segment equals a semicircle area minus zero triangle (since sin180° = 0), so segment = sector.

Diameter mode. When measuring a physical circle (pipe, wheel, hole), calipers measure diameter directly across the object. Radius requires locating the exact center, which introduces positioning error. Circumference measurement with a tape is prone to slippage and stretching. Area is almost never measured directly - it is computed.

The calculator clamps the angle to the range 0 - 360°. An angle of 360° yields a full circle: arc length equals circumference, sector area equals total area, and chord length returns to 0 (the chord endpoints coincide). Angles beyond one full revolution are geometrically redundant for a single-pass arc.

Yes, within JavaScript's 64-bit float range: approximately 5 × 10⁻³²⁴ to 1.8 × 10³⁰⁸. For astronomical scales (e.g., planetary orbits at 10¹¹ m), precision is about 1 mm. For atomic scales (10⁻¹⁰ m), you retain ~6 significant figures. The tool displays a warning if input approaches float limits.

Yes. The formula A_seg = 12r²(θ − sinθ) is valid for 0 < θ ≤ 2π. When θ > π, sinθ becomes negative, so the subtraction actually adds area - correctly producing the major segment. No special-case logic is needed.