What is arc length and what is the arc length formula?

Arc length is the distance along the curved perimeter of a circle or curve between two endpoints. For a circle of radius r and central angle theta in radians, the formula is s = r * theta. If the central angle is in degrees, the formula is s = 2 * pi * r * (theta / 360°) = (pi * r * theta) / 180°.

How do you calculate arc length from degrees vs radians?

In radians, arc length is simply the product of radius and angle: s = r * theta_rad. In degrees, because a full circle is 360° with circumference 2*pi*r, the arc length represents the fraction (theta / 360°) of the circumference: s = 2 * pi * r * (theta_deg / 360°) = (pi * r * theta_deg) / 180°. For example, a 90° arc on a circle of radius 10 cm has s = (pi * 10 * 90) / 180 = 5*pi ≈ 15.708 cm.

What is the arc length formula in calculus for curves?

For a differentiable function y = f(x) on the interval [a, b], the arc length is given by the definite integral: L = integral from a to b of sqrt(1 + [f'(x)]^2) dx. For parametric curves (x(t), y(t)) on [t1, t2], L = integral of sqrt((dx/dt)^2 + (dy/dt)^2) dt. In polar coordinates r = f(theta) on [alpha, beta], L = integral of sqrt(r^2 + (dr/dtheta)^2) dtheta.

How do you find arc length without the central angle (from chord and radius)?

If you know the chord length (c) and radius (r), first determine the central angle using trigonometry: theta = 2 * arcsin(c / (2*r)) in radians. Then substitute into the arc length formula: s = 2 * r * arcsin(c / (2*r)).

How do you find arc length from chord length and sagitta (arc height)?

Given chord length c and sagitta height h, calculate the circle radius using the intersecting chords theorem: r = (c^2 / (8*h)) + (h / 2). Then compute the central angle theta = 4 * arctan(2*h / c) in radians, and calculate arc length as s = r * theta.

What is the difference between an arc, a chord, a sector, and a segment of a circle?

An arc is the curved boundary portion along the circle perimeter. A chord is the straight line segment connecting the arc's two endpoints. A sector is the "pie-slice" region enclosed by the arc and two radii. A segment is the area bounded between the arc and its chord (A_segment = A_sector - A_triangle).

How do you calculate the area of a sector from arc length?

The area of a circular sector is given by A = (1/2) * r^2 * theta. Since s = r * theta, substituting s yields the elegant formula A = (1/2) * r * s.

What is the difference between a major arc and a minor arc?

A minor arc subtends a central angle less than 180° (theta < pi rad), representing less than half the circle's circumference. A major arc subtends an angle greater than 180° (180° < theta < 360°). An arc of exactly 180° (pi rad) is a semicircle (s = pi * r).

How is arc length used in real-world engineering, navigation, and construction?

Arc length is used extensively in civil engineering for curved highway and railway transitions, in architecture for arched doorways and vaulted ceilings, in mechanical engineering for belt-drive lengths and gear tooth pitches, and in terrestrial navigation (where 1 nautical mile = 1 arcminute of latitude along Earth's spherical surface).

How does this Arc Length Calculator work and what modes does it support?

The calculator features 5 flexible solver modes: solving arc length from radius and central angle (r + theta -> s), solving central angle from radius and arc length (r + s -> theta), solving from chord and radius (c + r -> s), solving from chord and sagitta height (c + h -> s), and calculus curve arc length integration (y = f(x)) with multi-unit conversions across metric and imperial dimensions.

100% Free • Circle & Calculus Arc Length Geometry Engine

Arc Length Calculator

Calculate circle arc length ($s = r\theta$), central angle, radius, chord length, sagitta height, circular sector area, and segment area. Includes calculus curve integration for $y = f(x)$, parametric, and polar curves.

Geometric Presets: Tap to load

Arc Length Workbench

Select calculation mode and configure geometric parameters.

s = r • θ

Solver Mode

Circle Radius ($r$)

Central Angle ($\theta$)

Key Geometric Ratios

Arc / Circumference 25.00%

Chord / Radius 1.414

Sagitta Height 2.929 m

Calculated Arc Length ($s$)

15.708 m

5π m • 25.00% of Circle Circumference (62.832 m)

Geometric Telemetry Matrix

📏 Arc Length ($s$)	15.708 m
📐 Central Angle ($\theta$)	90.000° (1.571 rad)
⭕ Circle Radius ($r$)	10.000 m (d = 20.00 m)
🎯 Chord Length ($c$)	14.142 m
🏹 Sagitta / Arc Height ($h$)	2.929 m
🍰 Sector Area ($A_{\text{sec}}$)	78.540 m²
🌙 Segment Area ($A_{\text{seg}}$)	28.540 m²
🔄 Circumference ($C$)	62.832 m

Why Use Our Arc Length Calculator? Real-World Problems It Solves

Whether you are a student solving trigonometry homework, a civil engineer designing curved roadway easements, or a fabricator bending sheet metal and conduit, finding circular dimensions is rarely a straightforward textbook problem. Our Arc Length Calculator eliminates common calculation roadblocks:

1. Inaccessible Center Points (Sagitta & Chord Mode):

In construction, surveying, and architectural restoration, finding the center of an existing arch or curved masonry wall is often physically impossible. By simply measuring the straight baseline span (chord length $c$) and the rise at the midpoint (sagitta $h$), our solver calculates the true circle radius, central angle, and outer perimeter arc length without guessing.

2. Multi-Directional Inverse Solving:

Standard calculators only work forward ($r, \theta \to s$). If you have a fixed material length (such as a 12-foot bent steel beam or a conveyor belt) and need to determine the required radius or angle of wrap, our tool performs instant inverse trigonometric calculations in both degrees and radians.

3. Automated Unit Conversion & Angle Formats:

Switch seamlessly between decimal degrees ($^\circ$), radians ($\text{rad}$), gradians ($\text{grad}$), arcminutes ($'$), and arcseconds ($''$). Radius dimensions convert automatically across metric (mm, cm, m, km) and imperial (in, ft, mi) standards.

4. Calculus Curve Integration on Demand:

For non-circular mathematical curves—such as hanging catenary power cables ($y = a\cosh(x/a)$), parabolic projectile trajectories ($y = x^2$), and trigonometric waveforms—our built-in numerical integrator evaluates $\int_a^b \sqrt{1 + [f'(x)]^2} dx$ instantly without manual integration tables.

How to Use the Arc Length Calculator (Step-by-Step Guide)

Step 1: Choose Solver Mode

Select the calculation mode matching your known parameters: Radius + Angle, Radius + Arc Length, Chord + Radius, Chord + Sagitta, or Calculus Integration.

Step 2: Enter Known Values

Input your numerical values and select your preferred units for radius (meters, feet, inches, cm) and angle (degrees, radians, gradians).

Step 3: Review Telemetry

Review the real-time geometric dashboard displaying arc length, chord length, sagitta height, sector area, and segment area.

Step 4: Copy or Export

Click Copy Arc Length Telemetry Card to copy a formatted calculation summary directly to your clipboard for CAD drawings, reports, or homework.

Key Features & Architectural Capabilities

5 Multidirectional Solvers

Solve forward from radius and angle, solve inverse for angle, reconstruct circles from chord and sagitta height, or integrate continuous calculus functions.

Circular Sector & Segment Suite

Automatically computes circular sector area ($A = \frac{1}{2} r s$) and circular segment area ($A_{\text{seg}} = A_{\text{sec}} - A_{\text{tri}}$) alongside linear arc dimensions.

100% Private In-Browser Math

All calculations execute locally via client-side JavaScript. Zero server roundtrips, zero latency, and complete privacy for proprietary engineering designs.

The Master Arc Length Formula Guide (Degrees, Radians & Gradians)

The arc length ($s$) of a circle represents the distance along the curved perimeter connecting two endpoints. In radian measure, the radian is fundamentally defined as the ratio of arc length to circle radius ($\theta = s/r$), producing the most elegant geometric relation: $$s = r \cdot \theta_{\text{radians}}$$

Core Circle Arc Length Formulations Across All Angle Units:

$$s = \frac{\pi \cdot r \cdot \theta^\circ}{180^\circ} = 2\pi r \left(\frac{\theta^\circ}{360^\circ}\right)$$ Arc Length Formula Degrees ($^\circ$)

$$s = r \cdot \theta_{\text{rad}}$$ Arc Length Formula Radians ($\text{rad}$)

$$s = \frac{\pi \cdot r \cdot \theta_{\text{grad}}}{200\,\text{grad}}$$ Arc Length Formula Gradians ($\text{grad}$)

Worked Step-by-Step Arc Length Examples

Example 1: Degrees Formula ($r = 14\,\text{cm}, \theta = 120^\circ$)

To find the arc length subtended by a $120^\circ$ angle on a circle of radius $14\,\text{cm}$: $$s = \frac{\pi \cdot 14 \cdot 120}{180} = \frac{28\pi}{3} \approx 29.322\,\text{cm}$$ The arc represents exactly $\frac{120}{360} = \frac{1}{3}$ of the circle's total circumference ($C = 2\pi(14) \approx 87.96\,\text{cm}$).

Example 2: Construction Arch ($c = 8\,\text{ft}, h = 2\,\text{ft}$)

For an arched doorway with span $c = 8\,\text{ft}$ and rise $h = 2\,\text{ft}$: $$r = \frac{8^2}{8(2)} + \frac{2}{2} = 4 + 1 = 5.000\,\text{ft}$$ $$\theta = 4\arctan\left(\frac{2(2)}{8}\right) = 4\arctan(0.5) \approx 1.8546\,\text{rad} \approx 106.26^\circ$$ $$s = 5.000 \times 1.8546 \approx 9.273\,\text{ft}$$

Master Benchmark Circle Arc Matrix Table (Standard Angles & Segments)

Exact trigonometric and geometric metrics for unit circle radius ($r = 1.0$).

Geometric Benchmarks

Arc Name	Angle (Degrees)	Arc Length ($s$)	Chord Length ($c$)	Sagitta ($h$)	Sector Area	Circumference %
Full Circle	$360^\circ$ ($2\pi\,\text{rad}$)	$2\pi \approx 6.283$	$0.000$	$2.000$	$\pi \approx 3.142$	$100.0\%$
Semicircle	$180^\circ$ ($\pi\,\text{rad}$)	$\pi \approx 3.142$	$2.000$	$1.000$	$\pi/2 \approx 1.571$	$50.0\%$
Golden Angle Arc	$137.508^\circ$	$2.400$	$1.864$	$0.635$	$1.200$	$38.2\%$
Quarter Circle	$90^\circ$ ($\pi/2\,\text{rad}$)	$\pi/2 \approx 1.571$	$\sqrt{2} \approx 1.414$	$1 - \sqrt{2}/2 \approx 0.293$	$\pi/4 \approx 0.785$	$25.0\%$
Equilateral Triangle Arc	$60^\circ$ ($\pi/3\,\text{rad}$)	$\pi/3 \approx 1.047$	$1.000$ ($c = r$)	$1 - \sqrt{3}/2 \approx 0.134$	$\pi/6 \approx 0.524$	$16.7\%$
Octagonal Arc	$45^\circ$ ($\pi/4\,\text{rad}$)	$\pi/4 \approx 0.785$	$\approx 0.765$	$\approx 0.076$	$\pi/8 \approx 0.393$	$12.5\%$

Arc Length Formula in Calculus (Cartesian, Parametric & Polar Curves)

1. Cartesian Explicit ($y = f(x)$) $$L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$$

Derived from summing infinitesimal Pythagorean segments $dL = \sqrt{dx^2 + dy^2} = \sqrt{1 + (dy/dx)^2} dx$.

2. Parametric Curves ($x(t), y(t)$) $$L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$$

Essential for cycloids, planetary orbits, and Bezier curves where $x$ and $y$ are parameterized by time or angle $t$.

3. Polar Coordinates ($r = f(\theta)$) $$L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta$$

Used for Archimedean and logarithmic spirals, cardioids, and rose curves in physics and antenna engineering.

Inverse Arc Length Solver: Reconstructing Radius, Chords & Sagitta

1. From Chord ($c$) and Radius ($r$):

Constructing an isosceles triangle from the circle center to chord endpoints gives $c = 2r \sin(\theta/2)$. The central angle is $\theta = 2\arcsin(c/2r)$, yielding: $$s = 2r \arcsin\left(\frac{c}{2r}\right)$$

2. From Chord ($c$) and Sagitta Height ($h$):

By the intersecting chords theorem, $h(2r - h) = (c/2)^2 \implies r = \frac{c^2}{8h} + \frac{h}{2}$. The central angle is $\theta = 4\arctan(2h/c)$, yielding: $$s = \left(\frac{c^2}{8h} + \frac{h}{2}\right) \times 4\arctan\left(\frac{2h}{c}\right)$$

Real-World Engineering, Navigation & Construction Applications

Everyday Engineering Implementations:

Civil Engineering & Road Design: Calculating horizontal highway curve transitions and railway track curvature lengths to manage centripetal forces.
Mechanical & Belt Drives: Computing contact wrap arc lengths on pulleys and gears: $s_{\text{wrap}} = r \theta_{\text{contact}}$.
Architecture & Carpentry: Determining sheet metal and timber lengths for curved arched barrel vaults, domes, and bridges.
Nautical Navigation: Defining the nautical mile as exactly $1\,\text{arcminute}$ of Earth's latitude ($s = R_{\text{Earth}} \times (\pi / 10800) \approx 1,852\,\text{m}$).

Frequently Asked Questions (FAQ)

Authoritative answers to common questions about circle arc lengths, central angles, chords, sagittas, and calculus curve integration.

Arc Name	Angle (Degrees)	Arc Length (\(s\))	Chord Length (\(c\))	Sagitta (\(h\))	Sector Area	Circumference %
Full Circle	\(360^\circ\) (\(2\pi\,\text{rad}\))	\(2\pi \approx 6.283\)	\(0.000\)	\(2.000\)	\(\pi \approx 3.142\)	\(100.0\%\)
Semicircle	\(180^\circ\) (\(\pi\,\text{rad}\))	\(\pi \approx 3.142\)	\(2.000\)	\(1.000\)	\(\pi/2 \approx 1.571\)	\(50.0\%\)
Golden Angle Arc	\(137.508^\circ\)	\(2.400\)	\(1.864\)	\(0.635\)	\(1.200\)	\(38.2\%\)
Quarter Circle	\(90^\circ\) (\(\pi/2\,\text{rad}\))	\(\pi/2 \approx 1.571\)	\(\sqrt{2} \approx 1.414\)	\(1 - \sqrt{2}/2 \approx 0.293\)	\(\pi/4 \approx 0.785\)	\(25.0\%\)
Equilateral Triangle Arc	\(60^\circ\) (\(\pi/3\,\text{rad}\))	\(\pi/3 \approx 1.047\)	\(1.000\) (\(c = r\))	\(1 - \sqrt{3}/2 \approx 0.134\)	\(\pi/6 \approx 0.524\)	\(16.7\%\)
Octagonal Arc	\(45^\circ\) (\(\pi/4\,\text{rad}\))	\(\pi/4 \approx 0.785\)	\(\approx 0.765\)	\(\approx 0.076\)	\(\pi/8 \approx 0.393\)	\(12.5\%\)

📏 Arc Length (\(s\))	15.708 m
📐 Central Angle (\(\theta\))	90.000° (1.571 rad)
⭕ Circle Radius (\(r\))	10.000 m (d = 20.00 m)
🎯 Chord Length (\(c\))	14.142 m
🏹 Sagitta / Arc Height (\(h\))	2.929 m
🍰 Sector Area (\(A_{\text{sec}}\))	78.540 m²
🌙 Segment Area (\(A_{\text{seg}}\))	28.540 m²
🔄 Circumference (\(C\))	62.832 m