Example 1
- Radius r: 10
≈ 62.83
What circumference at r = 10? ≈ 62.83; area ≈ 314.16.
Circle Circumference Calculator
Enter radius r or diameter d. The calculator computes circumference C = 2πr = πd. At r = 10 the result is about 62.83; at d = 10 about 31.42 (same as r = 5). The primary result is rim length — area A = πr² appears beside it as a helper.
Result
Enter radius r (in). Circumference appears here.
C = 2πr. Disk area πr² is on the circle area calculator. Sphere surface 4πr² has its own page.
How it works
r10
2πr20π
C≈ 62.83
Circle circumference, written C, is the length of the rim. The number π is defined as the ratio of that length to the diameter: π = C/d. The definition immediately yields C = πd. Because diameter and radius satisfy d = 2r, substitution gives C = π·(2r) = 2πr. Both forms describe the same Euclidean model — the classical circumference formula.
The calculator uses the number you enter and the Radius r / Diameter d switch. In radius mode it computes C = 2πr and reports d = 2r. In diameter mode it computes C = πd and reports r = d/2. At r = 10 the result is about 62.83 in the same length unit. At d = 10 the result is about 31.42 — the same as 2πr at r = 5. Comma and period in the field are equivalent.
The domain is r > 0 (or d > 0). Zero and negative values do not describe a real rim length; a blank field is an error, not circumference 0. Disk area A = πr² appears beside the result as a check from the same r, but it does not replace the circle-area page. Sphere surface 4πr² and arc length of a sector are separate models.
Historically, Archimedes in Measurement of a Circle (3rd century BCE) bounded the circumference-to-diameter ratio by comparing perimeters of inscribed and circumscribed regular polygons. For 96-gons he obtained 3 10/71 < C/d < 3 1/7, about 3.1408 < π < 3.1429. Those are strict geometric bounds, not a tape-measure reading. Modern π is a transcendental constant; NIST lists it among mathematical constants, and the browser uses its built-in constant π.
Worked numbers on this page: r = 10 → C ≈ 62.83; r = 1 → C ≈ 6.28; d = 10 → C ≈ 31.42; r = 2 → C ≈ 12.57; r = 7 → C ≈ 43.98; r = 0.5 → C ≈ 3.14; r = 12 → C ≈ 75.40; r = 3 → C ≈ 18.85. The Pythagorean theorem does not compute this circumference: there are no sides a and b here, only one radius or one diameter. The result is a length, not an angle in radians.
Formula
C = 2 × π × r and equivalently C = π × d, where d = 2r.
Symbol C is circumference (rim length). Symbol r is the radius — distance from center to the circle. Symbol d is the diameter. Constant π is by definition the ratio C/d. Derivation: π = C/d ⇒ C = πd; d = 2r ⇒ C = 2πr.
Mathematical model: classical circle circumference (Euclidean geometry), with Archimedes’ historical bounds on C/d in Measurement of a Circle.
Theory, definition of π, and units
Definition
The circumference of a circle of radius r is C = 2πr. Equivalently, with diameter d = 2r, one has C = πd. The constant π is the ratio of circumference to diameter in Euclidean plane geometry: π = C/d for every circle. This calculator accepts r or d (according to the form fields) and returns C from that definition.
A circle (as a curve) is the set of plane points at fixed distance r from a centre; a disk is the filled region. Circumference is the arc length of the boundary. In analysis the length follows from ∫√(x′²+y′²) dt for (r cos t, r sin t), t ∈ [0, 2π], which equals 2πr.
Historical and conceptual context
In Measurement of a Circle Archimedes proved that the circumference exceeds every inscribed regular polygon and is less than every circumscribed one, and that 3·(10/71) < π < 3·(1/7), i.e. 223/71 < π < 22/7, by doubling sides up to a 96-gon. Euclid’s Elements (Books III, XII) develop circle theory and similarity without modern π notation.
Later bounds (Liu Hui, Zu Chongzhi, Viète, Wallis, Machin) sharpened π. Conceptually C = 2πr is not a school approximation — it is the Euclidean length once π is fixed. On a sphere or in hyperbolic geometry the circumference-to-diameter ratio is not the constant π.
Formal properties
Domain: r > 0 (or d > 0). The map C(r) = 2πr is linear, continuous, and strictly increasing. Homogeneity: under length scaling by λ > 0, circumference scales as λ¹. Link to area: A = πr² = C²/(4π), so C determines the disk area without a separate radius input.
Identities: C = 2πr = πd; r = C/(2π); d = C/π. Arc length for central angle θ in radians is s = rθ = (C/(2π))·θ. A full turn θ = 2π recovers C.
Isoperimetry in the plane: among closed curves of fixed perimeter, the circle maximises area (the isoperimetric inequality; classical treatments via Weierstrass and the calculus of variations).
Full derivation and Archimedean bounds
Step 1 (parametrisation). Let x = r cos t, y = r sin t. Then √(x′²+y′²) = r, and integrating from 0 to 2π yields C = 2πr.
Step 2 (similarity). All circles are similar; the ratio C/d is therefore a universal constant, denoted π.
Step 3 (Archimedes). The perimeter of a regular n-gon inscribed in a circle of diameter 1 is less than C; the circumscribed n-gon is greater. For n = 96 Archimedes obtained 223/71 < π < 22/7. Modern definitions of π (series, integrals) recover the same constant to arbitrary precision.
Step 4 (area link). Archimedes also showed that the area of a disk equals that of a right triangle with legs r and C, hence A = (1/2)rC = πr².
Special cases
For r = 1 (unit circle) C = 2π ≈ 6.283185…. For d = 1 one has C = π ≈ 3.141592…. School approximations C ≈ 3.14·d or (22/7)·d are useful by hand but are not the exact value of π.
As r → 0, C → 0. Ellipse perimeter is not 2πr — it requires a complete elliptic integral of the second kind; this calculator treats only circles.
Numerical and computational notes
The browser uses π (about 15–17 significant digits in floating-point arithmetic). For ordinary radii the representation error in π is negligible next to tape-measure error. The product 2·π·r is numerically stable.
Non-positive input is rejected. Comma and period are equivalent. The calculator does not return a partial arc length for an arbitrary angle — that belongs to a separate arc-length tool if present on the site.
Link to area and cylinder volume: knowing C determines r = C/(2π), then area A = C²/(4π) and — given a height — cylinder volume V = C²h/(4π). Those steps are purely algebraic and need no separate radius measurement if the girth is correct. A relative error in C enters A and V with weight 2 because C appears squared.
In differential geometry the circumference is a special case of the length of a regular curve. Circle curvature is 1/r; total curvature around a simple closed convex curve equals 2π by the rotation index of the tangent. Those facts do not change C = 2πr, but place it in the broader theory of plane curves.
Practical approximations: 22/7 overestimates π by about 0.04%; 355/113 (Zu Chongzhi’s Milü) is accurate to six decimal places. The calculator does not switch to rationals — it uses the browser’s built-in floating-point constant. For millimetre-tolerance drafting the gap between π and 3.1416 is often negligible next to measurement error.
Transcendental character: π is irrational (Lambert) and transcendental (Lindemann–Weierstrass), so C/d cannot equal a ratio of integers for a perfect Euclidean circle. Finite decimal displays are truncations of that constant, not a different geometric law.
Related concepts and authorities
Related models: circle area, arc length, circumference from diameter, and cylinder volume from circumference (which uses C = 2πr). Circumference alone does not fix the centre’s location in the plane.
Authorities: Archimedes, Measurement of a Circle (Heath); Euclid, Elements, Books III and XII; NIST DLMF §3.12 (π); MathWorld — Circle; Wikipedia — Pi; Wikipedia — Measurement of a Circle. The calculator does not replace a course in analysis or history of mathematics.
How to use
- Choose Input: Radius r or Diameter d.
- Enter a positive length, for example 10. Use a comma or a period.
- Click Calculate or wait for the live result. r = 10 gives about 62.83. d = 10 gives about 31.42.
- Check the panel: C, r, d, the πd check, area πr², and the circle diagram.
- Paint coverage on the disk belongs on the circle-area page. Sphere surface has its own calculator.
Symbol glossary
C = 2πr = πd. At r = 10, C ≈ 62.83. π = C/d by definition. Domain: r > 0.
- C
- Circumference — rim length. r = 10 → ≈ 62.83. d = 10 → ≈ 31.42.
- r
- Radius from center to the circle. Default input. r = d/2.
- d
- Diameter, d = 2r. Option in the Input switch. C = πd.
- π
- Ratio C/d. Here π. Archimedes: 3 10/71 < π < 3 1/7.
Examples
Example 2
- Radius r: 1
≈ 6.28
What circumference at r = 1? ≈ 6.28 (2π, not bare π).
Example 3
- Diameter d: 10
≈ 31.42
What circumference at d = 10? ≈ 31.42 — same as 2πr at r = 5.
Example 4
- Radius r: 2
≈ 12.57
What circumference at r = 2? ≈ 12.57 (4π — not area).
Example 5
- Radius r: 7
≈ 43.98
What circumference at r = 7? ≈ 43.98.
Example 6
- Radius r: 0.5
≈ 3.14
What circumference at r = 0.5? ≈ 3.14 — circumference, not area at r = 1.
Example 7
- Radius r: 12
≈ 75.40
What circumference at r = 12? ≈ 75.40.
Example 8
- Radius r: 3
≈ 18.85
What circumference at r = 3? ≈ 18.85.
Related calculators
Frequently asked questions
How is circumference different from circle area?
Circumference C = 2πr is the rim length. Area A = πr² is the disk measure. At r = 10 in the circumference is about 62.83 in and the area about 314.16 in². Two formulas; full area lives on the circle-area calculator.
How is π defined?
π is the ratio of a circle’s circumference to its diameter: π = C/d. That immediately gives C = πd. Because d = 2r, you also have C = 2πr. In the browser this calculator uses π.
Can I enter diameter instead of radius?
Yes. In Input choose Diameter d. Then the calculator computes C = πd and shows r = d/2. Example: d = 10 in gives C ≈ 31.42 in — the same as 2πr at r = 5 in.
Why does the domain require r > 0?
Radius and diameter describe a real circle only for positive values. Zero is not a meaningful rim; a negative length is not a length. A blank field is an error, not circumference 0.
What is the circumference at r = 10?
C = 2π·10 ≈ 62.83 in the same unit as r. That is the first example on this page. Area from the same r is about 314.16 square units.
Is C = πd the same as 2πr?
Yes. From π = C/d you get C = πd. Substituting d = 2r yields C = π·2r = 2πr. The result panel shows both paths as a check.
Do the currency and unit settings in the page header affect the result?
No. The unit switch only changes the cm or in label; it does not multiply the entered digits. If you measured 10 cm and want inches, convert the length yourself first.
What about Archimedes’ bounds?
In Measurement of a Circle Archimedes proved that the circumference-to-diameter ratio lies between 3 10/71 and 3 1/7, about 3.1408 < π < 3.1429. Those are historical polygon bounds, not a substitute for π here.
Is the result in radians?
No. Circumference is a length in in or cm. Radians measure angles — they belong on the degrees-and-radians page, not here.
Is sphere surface area on this page?
No. Sphere surface is 4πr² and has its own calculator. At r = 10 it is about 1256.64 square units, not 62.83 of rim length.
How do I calculate circle circumference with C = 2πr?
Multiply the radius by 2π: C = 2πr. At r = 10: C ≈ 62.83. At r = 5: C ≈ 31.42. From diameter the shorter form is C = πd — at d = 10 that same result ≈ 31.42.
How do I find circumference from diameter?
Choose input “Diameter d” and enter d. The calculator computes C = πd and shows r = d/2. Example: d = 10 gives C ≈ 31.42 — the same as 2πr at r = 5.
How does this differ from the circle area calculator?
Here the primary result is C = 2πr (rim length). Area A = πr² appears beside it for the same r. Starting from area alone, or reversing from C to r, lives on other cards when you need that path.
How do I find radius when I only know the circumference?
In theory r = C / (2π) and d = C / π. This page goes the other way — from r or d to C. If you measured the rim and need r, rearrange on paper or use a tool that accepts C as input.
What is the circumference of a circle with radius 5?
C = 2π·5 ≈ 31.42 in the same unit as r. The same value comes from d = 10 via πd. The diameter-10 example on this page fills exactly that layout.
Sources
The definition of π, formulas C = 2πr = πd, and the historical note below come from established references.
- Wikipedia — Measurement of a Circle (Archimedes)
- NIST — CODATA / Mathematical constants (π)
- MathWorld — Circle
- MathWorld — Pi
- Wikipedia — Circumference
- Wikipedia — Pi
Page Updated November 2, 2026.