Arc Length and Sector Calculator: Arc, Sector Area and Chord

About this calculator

The arc length and sector calculator takes the radius $r$ of a circle and a central angle $\theta$ (entered in degrees) and returns three things: the length $L$ of the arc, the area $A$ of the sector (the “pizza slice” bounded by the centre and the arc), and the length of the chord joining the ends of the arc. Conversely, given the arc length and the radius, it finds the central angle.

Anything that is “part of a circle” can use it: the distance swept by a clock hand, the length of a bend in a road, the area of a slice of pizza, the angle corresponding to one tooth pitch on a gear, how many kilometres 1° along a meridian is on the Earth’s surface. It is also where the two angle units, degrees and radians, are genuinely connected—a radian is defined as the ratio of arc length to radius.

What it does not do: it does not compute the circumference and area of the whole circle (enter 360° to get them); it does not compute the area of a circular segment (the region between arc and chord)—but this page gives the formula, sector area minus triangle area; it does not normalise angles to 0°–360°; and it does not convert units.

Formula and variables

First convert the angle from degrees to radians, $\theta = \theta_{\deg} \times \pi / 180$ , then

L = r\,\theta,\qquad A = \tfrac{1}{2}\,r^2\theta,\qquad \text{chord} = 2r\sin\frac{\theta}{2}

Solving the angle from the arc length: $\theta_{\deg} = \dfrac{L}{r} \times \dfrac{180}{\pi}$ . The area of the circular segment (between arc and chord) is not an output, but it is the sector minus the isosceles triangle:

A_{\text{segment}} = \tfrac{1}{2}\,r^2\,(\theta - \sin\theta)

Symbol	Meaning	Calculator field	Unit	Range and notes
$r$	Radius of the circle	Radius r	length	$r > 0$ ; default 10
$\theta_{\deg}$	Central angle (degrees)	Angle (°)	degrees	Usually $0 < \theta_{\deg} \le 360$ ; default 60; the engine does not limit the range—more than 360° counts as “more than a full turn”
$\theta$	Central angle (radians)	(intermediate)	rad	$\theta = \theta_{\deg}\,\pi/180$ ; 60° = π/3 ≈ 1.0472 rad
$L$	Arc length	Arc length	length	Output; equals the circumference $2\pi r$ when $\theta_{\deg} = 360$ ; can be entered to solve for the angle
$A$	Sector area	Sector area	length²	Output; equals the circle’s area $\pi r^2$ when $\theta_{\deg} = 360$
chord	Straight-line distance between the ends of the arc	Chord length	length	Output; equals the diameter $2r$ when $\theta_{\deg} = 180$ , its maximum

Radius, arc length and chord share one length unit; the area is in the square of that unit.

How to use this calculator

Open the panelthis page corresponds to “Math tools → Formulas → Arc length / sector” (/#/mathtools/formula?calculator=calculator.arc-sector). It opens with radius r = 10 and angle 60° filled in, and the arc length, sector area and chord computed.
Compute arc length, area and chordchange the radius and angle (the angle in degrees, not radians); the three fields marked “computed” update live to at most 8 decimal places with trailing zeros removed.
Solve the angle from the arc lengthclear “Angle (°)”, enter the known arc length in “Arc length” and keep the radius; the angle field switches to “computed” and shows degrees. The sector area and chord then show 0 (the engine treats the unknown angle as 0); ignore them, or enter the solved angle back into the angle field to read them.
Full circle and semicircleenter 360 for the circumference and the circle’s area (the chord is theoretically 0; the interface may show floating-point noise such as 1.469576e-15, which means 0); enter 180 for the semicircular arc, the semicircle’s area and the diameter.
Segment areasubtract the triangle area $\tfrac12 r^2 \sin\theta$ from the sector area output, or substitute directly into $\tfrac12 r^2(\theta - \sin\theta)$ .

Worked examples

All four examples were recomputed by the engine’s arc-sector compute routine; the display convention is at most 8 decimal places with trailing zeros removed.

Example 1: radius 6, central angle 60°

Radians $\theta = 60 \times \pi/180 = \pi/3 \approx 1.04719755$ .
Arc length $L = 6 \times \pi/3 = 2\pi \approx 6.28318531$ .
Sector area $A = \tfrac12 \times 36 \times \pi/3 = 6\pi \approx 18.84955592$ .
Chord $= 2 \times 6 \times \sin 30° = 12 \times 0.5 = 6$ .

Interface shows: arc length 6.28318531, sector area 18.84955592, chord 6. At a central angle of 60° the two radii and the chord form an equilateral triangle, so the chord is exactly equal to the radius.

Figure 1: Example 1. Arc length and area both need the angle in radians; the chord uses 2r·sin(θ/2), where θ/2 may be in degrees or radians as long as it matches the sine’s convention

Example 2: panel defaults, radius 10, 60°

$L = 10 \times \pi/3 \approx 10.47197551$ .
$A = \tfrac12 \times 100 \times \pi/3 \approx 52.35987756$ .
Chord $= 20 \sin 30° = 10$ .

Interface shows: 10.47197551, 52.35987756, 10. Compared with Example 1 the radius is scaled by $10/6$ ; arc length and chord scale by the same factor, the area by $(10/6)^2 = 2.78$ .

Example 3: a major arc—radius 1.5, 270°

$\theta = 270 \times \pi/180 = 3\pi/2 \approx 4.71238898$ .
$L = 1.5 \times 3\pi/2 \approx 7.06858347$ (three quarters of the circumference).
$A = \tfrac12 \times 2.25 \times 3\pi/2 \approx 5.3014376$ (three quarters of the circle’s area).
Chord $= 3 \sin 135° = 3 \times 0.70710678 \approx 2.12132034$ .

Interface shows: 7.06858347, 5.3014376, 2.12132034. Beyond 180° the chord starts to shrink again—the 270° chord is as long as the 90° chord, because they are the two sides of the same chord.

Example 4: solving the angle from the arc length

What central angle does an arc of length 5 on a circle of radius 10 subtend? Clear the angle, enter arc length 5 and radius 10.

$\theta = L/r = 0.5$ rad.
$\theta_{\deg} = 0.5 \times 180/\pi \approx 28.64788976°$ .

Interface shows: angle 28.64788976. The sector area and chord fields show 0 here, which is the solve-mode convention (see Assumptions). Enter 28.64788976 back into the angle field to get sector area 25 and chord 4.94808114.

Fix the radius at 6 and sweep the central angle from 0° to 360°: the arc length grows linearly with the angle (0 → 37.70), while the chord first grows and then shrinks, reaching its maximum of 12 (the diameter) at 180° and returning to 0 at 360°. For small angles the two curves almost coincide—the ratio of chord to arc, $\sin(\theta/2)/(\theta/2)$ , is 0.9987 at 10°, 0.9549 at 60° and 0.6366 at 180°.

Principles and derivation

Arc length: a fraction of the circumference

The whole circumference $2\pi r$ corresponds to a central angle of 360°; an arc of central angle $\theta_{\deg}$ is the fraction $\theta_{\deg}/360$ of the full circle, so

L = \frac{\theta_{\deg}}{360} \times 2\pi r = r \times \left(\theta_{\deg} \times \frac{\pi}{180}\right) = r\,\theta

The quantity in brackets is the angle in radians. Definition of the radian: the ratio of arc length to radius, $\theta = L/r$ . One radian is the central angle whose arc is exactly as long as the radius, about 57.2958°; the full circle is $2\pi$ radians. Written in radians, $L = r\theta$ contains no constant at all—which is why mathematics and physics almost always use radians.

Sector area: a fraction of the circle’s area

By the same proportion: $A = \dfrac{\theta_{\deg}}{360}\,\pi r^2 = \tfrac12 r^2 \theta$ . Another view slices the sector into infinitely many thin triangles, each with a small arc $r\,d\theta$ as base and $r$ as height, area $\tfrac12 r \cdot r\,d\theta$ ; summed, they give $\tfrac12 r^2 \theta$ . This is the same as $A = \tfrac12 L r$ (“base × height / 2” with the arc as base and the radius as height).

Chord: an isosceles triangle

The centre and the two ends of the arc form an isosceles triangle with legs $r$ and apex angle $\theta$ . Drop a perpendicular from the centre to the chord, splitting it into two right triangles, each with half the chord as opposite side, $r$ as hypotenuse and $\theta/2$ as apex angle; hence half the chord $= r\sin(\theta/2)$ and the chord $= 2r\sin(\theta/2)$ . At $\theta = 180°$ , $\sin 90° = 1$ and the chord $= 2r$ is the diameter, the maximum possible chord; beyond that the chord shrinks again, and at 360° the ends coincide and the chord is 0.

Segment area

The region bounded by the arc and the chord is a circular segment. It equals the sector minus the isosceles triangle: the triangle’s area is $\tfrac12 r^2 \sin\theta$ (the two-sides-and-included-angle formula), so $A_{\text{segment}} = \tfrac12 r^2(\theta - \sin\theta)$ . In Example 1 this is $18.84955592 - 15.58845727 = 3.26109865$ . The cross-sectional area of liquid in a horizontal cylindrical tank and the area above an arched doorway both use this expression.

Small-angle approximation

For small $\theta$ , $\sin\theta \approx \theta$ (in radians), so chord ≈ arc, and the segment area ≈ $\tfrac12 r^2 \cdot \theta^3/6$ tends to 0. This underlies the “chord for arc” substitution in surveying and astronomy: a 1° arc and its chord differ by only 0.0013%, and on the Earth the 1° arc of 111.195 km and chord of 111.194 km are practically indistinguishable.

Why a full turn is 360°

The 360 comes from Babylonian base-60 arithmetic and a rough count of about 360 days in a year; it has 24 divisors and is convenient to subdivide, but it is a human convention, not a geometric fact. Radians are fixed by the circle’s own proportions: a full turn is $2\pi$ . The name “radian” was proposed by James Thomson in 1873, though the concept goes back to Roger Cotes (1714).

Assumptions

The central angle is entered in degrees: the engine multiplies by $\pi/180$ internally; if your angle is already in radians, multiply by $180/\pi$ before entering it.
Angles are not normalised: entering 400° is computed as “one turn plus 40°”, arc length and area keep growing, and the chord is computed from $\sin 200°$ as a negative value; negative angles give negative arc lengths and areas. The meaningful range is $0° < \theta \le 360°$ .
Positive radius: $r > 0$ . Solving the angle with $r = 0$ divides by zero.
Solving affects only the angle field: when solving the angle from the arc length, the sector area and chord are computed with “angle = 0” and show 0; ignore them.
One length unit: radius, arc and chord share a unit; the area is its square.

Scope and limitations

Can calculate: arc length, sector area and chord for any positive radius and any central angle (degrees); the central angle from arc length and radius. Results to at most 8 decimal places.
Approximate only: the chord of a full circle (360°) is theoretically 0; the interface may show floating-point noise such as 1.469576e-15—read it as 0.
Cannot calculate: segment area (use the formula on this page by hand), the angle or radius from chord length or area, elliptical arcs, the length of spirals, or arcs on curved surfaces in 3D.
Does not: restrict the angle to 0°–360°, accept degrees-minutes-seconds or radians as input, or convert units.
Numerics: for extremely large radii combined with extremely small angles the relative precision of $\sin(\theta/2)$ degrades, with no effect at everyday magnitudes.

Common mistakes

Entering radians in “Angle (°)”: entering 1.0472 gives 1.0472°, and the arc length is only 0.1097 (radius 6) instead of 2π. The field takes degrees.
Forgetting to convert to radians by hand: $L = r\theta$ holds only in radians; multiplying the radius by 60 directly gives 360, not 6.28.
Taking the chord for the arc: the two are close for small angles but very different for large ones—at 180° the chord is 12 while the arc is 18.85 (radius 6).
Angles over 360°: the engine does not wrap around, and the chord for 400° is negative. Subtract whole turns first.
Reading area and chord in solve mode: they show 0 as a matter of convention, not geometry.
Scaling the area proportionally: doubling the radius doubles arc length and chord, but the sector area becomes 4 times as large.
Floating-point noise in the full-circle chord: 1.469576e-15 is 0, not a “very small chord”.

Typical use cases

Teaching: what a radian really is

Enter radius 1 and angle 57.29577951 (one radian): the arc length is exactly 1. Then enter 360 and watch the arc length become 6.28318531 = 2π. A radian is “measuring the arc with the radius as your ruler”.

Sharing pizza and cake fairly

A 30 cm diameter pizza cut into 8 slices: r = 15, angle 45°; each slice has area 88.35729338 cm², crust arc 11.78097245 cm and chord between the straight edges 11.4805 cm. Exactly one eighth of the whole 706.86 cm².

Bends, arches and clocks

The radius and turning angle of a road bend give the bend’s length; the radius and opening angle of an arch give the arc length (material) and chord (span); the hour hand turns 30° per hour and the minute hand 120° in 20 minutes, and with the hand length you get the distance travelled by the tip.

The Earth and navigation

With r = 6371 km, the 1.8532 km arc for one arcminute (1/60°) is the origin of the nautical mile; 1° of latitude along a meridian ≈ 111.19 km. In engineering and surveying the error of substituting chord for arc is negligible at small angles.

FAQ

Should I enter degrees or radians?

Degrees. The engine multiplies by $\pi/180$ internally. If you know the angle in radians, multiply by $180/\pi$ (about 57.2958) before entering.

How do I get the circumference and area of the whole circle?

Enter 360 for the angle: the arc length is the circumference $2\pi r$ and the sector area is the circle’s area $\pi r^2$ . The chord is theoretically 0 and the interface may show tiny floating-point noise.

How do I find the area of the segment between arc and chord?

Sector area minus triangle area: $\tfrac12 r^2(\theta - \sin\theta)$ with $\theta$ in radians. Example 1 gives 3.2611.

Why is the chord for 270° the same as for 90°?

It is the same chord: a chord divides the circle into two arcs, and the 90° minor arc and the 270° major arc share it, since $\sin 135° = \sin 45°$ .

Why do the area and chord become 0 when I solve for the angle?

In solve mode the engine treats the unknown angle as 0 when computing the other outputs, producing meaningless values. Enter the solved angle back into the angle field to see the correct area and chord.

Can I solve for the angle from the chord?

Not with this calculator (it solves only from the arc length). By hand: $\theta = 2\arcsin\big(\text{chord}/(2r)\big)$ , noting that the chord cannot exceed the diameter.

References and further reading

The arc-sector definition in the CalcX engine source src/data/formulas.ts (angle entered in degrees and converted to radians internally; targets arcLen, sectorArea, chordLen, and the angle solved from arc length); every figure on this page was recomputed by that engine.
Wikipedia, Circular sector （访问日期：2026-09-08）—sector area and arc length formulas.
Wikipedia, Arc length （访问日期：2026-09-08）—circular arc length and the arc length of general curves.
Wikipedia, Radian （访问日期：2026-09-08）—definition, history and conversion of the radian.
Wikipedia, Circular segment （访问日期：2026-09-08）—segment area and chord formulas.

Privacy

All inputs and calculations run inside your browser and are never uploaded to a server. The text on this page is static content; it neither contains nor records any user input.

Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

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