Mathematics · Geometry
Arc Length Calculator
Calculate circle arc length from radius and central angle.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Change an input to reshape this diagram.
Calculation steps
- 60/360×37.69911184307752=6.283185307179586.
- Report Arc length=6.283185307179586, Circumference=37.69911184307752.
Understand Arc length
One idea, three depths
Choose how deeply to explain Arc length
Calculate circle arc length from radius and central angle.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Arc length to answer this question: calculate circle arc length from radius and central angle? Enter Radius r and Central angle θ; the calculator shows Arc length. For example: A 60° arc on radius 6 has length 2π. The answer tells you Arc length.
Age 15Explain it to a 15-year-oldConnect it to the formula
Arc length is the angle's fraction of the full circumference. The rule is s=(θ/360°)2πr. Its input values are Radius r, Central angle θ (°), and the main result is Arc length. For example: A 60° arc on radius 6 has length 2π.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated arc length relation over the valid real-number domain stated below. The implemented relation is s=(θ/360°)2πr, evaluated from Radius r, Central angle θ (°) to produce Arc length. Arc length is the angle's fraction of the full circumference. Use the central angle, not an inscribed angle.
Inputs and valid domain
- Radius r must be a finite real number, at least 0.
- Central angle θ must be a finite real number in °.
Important boundary: Use the central angle, not an inscribed angle.
The formula
s=(θ/360°)2πr
How the calculator works through it
It substitutes Radius r, Central angle θ into the formula and exposes every numerical step above. The main output is Arc length, accompanied by Circumference.
Read the result correctly
The Arc length is the direct answer to “calculate circle arc length from radius and central angle.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
A 60° arc on radius 6 has length 2π.
Where this model stops being reliable
Use the central angle, not an inscribed angle.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Arc length works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Arc length uses s=(θ/360°)2πr. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Ratios between measured quantities
Ratios help you check the scale, units and proportional meaning of Arc length.
Review this foundation about 4 min
Optional enrichment
- Angles and geometric relationships
Angle language provides useful geometric context for extending Arc length to related shapes and constructions.
Review this foundation about 4 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Radius r, Central angle θ.
- Evaluate the principal relationship: s=(θ/360°)2πr.
- Return Arc length and check the domain conditions described above.
Python
from math import *
def arc_length(a, b) -> float:
return ((b / 360.0) * (2.0 * (pi * a)))
assert abs(arc_length(6, 60) - 6.283185307179586) < 1e-6 * max(1.0, abs(6.283185307179586))
C
#include <assert.h>
#include <math.h>
double arc_length(double a, double b) {
return ((b / 360.0) * (2.0 * (3.141592653589793 * a)));
}
int main(void) {
const double expected = 6.283185307179586;
const double actual = arc_length(6, 60);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double arc_length(double a, double b) {
return ((b / 360.0) * (2.0 * (std::numbers::pi * a)));
}
int main() {
constexpr double expected = 6.283185307179586;
const double actual = arc_length(6, 60);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double arc_length(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global arc_length
section .text
arc_length:
push rbp
mov rbp, rsp
sub rsp, 80
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
mov rax, 0x4076800000000000
movq xmm0, rax
movsd [rbp-40], xmm0
movsd xmm0, [rbp-16]
divsd xmm0, [rbp-40]
movsd [rbp-32], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-56], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-72], xmm0
movsd xmm0, [rbp-72]
mulsd xmm0, [rbp-8]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-56]
mulsd xmm0, [rbp-64]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-32]
mulsd xmm0, [rbp-48]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = arc_length(a, b)
result = ((b / 360.0) * (2.0 * (pi * a)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := ((b / 360.0) * (2.0 * (Pi * a)));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Arc Length Calculator. MW SysArc Tools. https://math.mwsysarc.com/geometry/circle-arc-length
MLA 9
MW SysArc. “Arc Length Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/geometry/circle-arc-length. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Arc Length Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/geometry/circle-arc-length.
Harvard
MW SysArc (2026) ‘Arc Length Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/geometry/circle-arc-length (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_arc_length_2026,
author = {{MW SysArc}},
title = {Arc Length Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/geometry/circle-arc-length},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Arc Length Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/geometry/circle-arc-length
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Arc length do?
Calculate circle arc length from radius and central angle.
How does the Arc length work?
The calculator applies s=(θ/360°)2πr. Arc length is the angle's fraction of the full circumference.
What can I learn from the Arc length?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .