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Arc Length Calculator

Find the length of a circular arc from the radius and central angle, chord or sector area, plus the sector's area and perimeter.

Level: Grades 10–11

  • Common Core standard HSG.C.B.5Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Printable worksheet

Arc Length Calculator

Know the answer you need? Pick an input to work backward.

Inputs
I know the radius and the

The angle at the center between the two radii, up to one full turn (360°).

Result

Arc length (s)

10.472

An arc of radius 10 spanning 60° is 10.472 long; its sector covers 52.3599 square units.

Central angle
60 °
Central angle
1.0472 rad
Sector area
52.3599 square units
Sector perimeter
30.472
Chord length
10
Share of the full circle
16.67%

See the step-by-step solution ↓

Step-by-step solution

  1. Arc length needs the angle in radians: multiply degrees by π/180

    θ=60×π180=1.047198

  2. Arc length is the radius times the angle in radians: s = rθ

    s=10×1.047198=10.471976

  3. Sector area is half the radius squared times the angle, which is the slice's share of the whole circle

    A=12×102×1.047198=52.359878

  4. The sector's perimeter adds the two radii to the arc

    P=2×10+10.471976=30.471976

  5. Find the straight-line chord between the arc's ends using half the angle, since half the chord forms a right triangle

    c=2×10×sin 30°=10

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find the length of a curved piece of a circle, plus the area of the pie slice it makes. An arc is a piece of a circle's edge. The sector is the pie slice between the arc and two radii (lines from the center to the edge). The arc's length depends on two things: the circle's radius and how wide the slice opens at the center, called the central angle. With the angle in radians, the rule is short and friendly: arc length equals radius times angle.

Give the radius plus one more value: the central angle, the chord (the straight line between the arc's ends) or the sector's area. The calculator then works out the angle, the arc length, the sector area, the sector perimeter and the chord. Angles can be entered in degrees, radians, gradians or turns.

Where it's used

Why this math matters outside the classroom:

  • Running tracks. The curved end of a track with a 36.5 m inner radius is a half circle, so its inside edge is π × 36.5, about 114.7 m long.
  • Pizza crust. A 90° slice from a pizza with a 15 cm radius has an outer crust (π/2) × 15, about 23.6 cm long.
  • Road curves. Highway engineers find that a curve with a 400 m radius turning through 30° covers 400 × π/6, about 209 m of road.
  • Clock hands. The tip of a 10 cm minute hand travels 2π × 10 × (20/60), about 20.9 cm, in 20 minutes.
  • Sprinkler coverage. A sprinkler that sweeps 120° with a 6 m reach waters a sector of ½ × 6² × (2π/3), about 37.7 m².

Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSG.C.B.5. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose what you know besides the radius: the central angle, the chord length or the sector area.
  2. Enter the radius and that second value. Use one length unit throughout; an area must be in the matching square unit.
  3. If you entered an angle, pick its unit. Angles from just above 0 up to one full turn are accepted.
  4. Read the arc length at the top and the angle, sector area, perimeter and chord below it.

What the inputs mean

Central angle (θ)
The angle at the circle's center between the radii to the two ends of the arc. 360° (2π radians) is the whole circle.
Chord length (c)
The straight-line distance between the ends of the arc. A chord fits two arcs, a short one and a long one; this calculator gives the shorter, which never exceeds a semicircle.
Sector area (A)
The area of the pie-slice shape enclosed by the arc and the two radii.

Formula and method

θ (radians) = θ (degrees) × π ÷ 180
s = r × θ
A = ½ × r² × θ
P = 2r + s
c = 2r × sin(θ ÷ 2)
from a chord: θ = 2 × arcsin(c ÷ 2r);  from an area: θ = 2A ÷ r²

In these formulas r is the radius and θ ("theta") is the central angle. s is the arc length, A is the sector area, P is the sector perimeter and c is the chord. π ("pi") is about 3.14159. sin is the sine function, and arcsin runs it in reverse to turn a ratio back into an angle.

A radian is the angle whose arc is exactly one radius long. So an angle of θ radians cuts off an arc θ radii long. That is why s = rθ needs the angle in radians; with degrees you first multiply by π/180. The sector area follows the same idea. The slice is θ ÷ 2π of the whole circle's area πr², which simplifies to ½r²θ.

The chord is the base of a triangle whose other two sides are both the radius. Splitting it down the middle gives two right triangles, each with half the angle, θ/2. So half the chord is r sin(θ/2). Running that relation backwards turns a known chord into the central angle.

Worked example

A curved garden path follows a circle of radius 12 m and turns through 75°. How long is it, and how much lawn does the slice inside it cover?

  1. Convert the angle: θ = 75 × π ÷ 180 ≈ 1.309 rad.
  2. Arc length: s = 12 × 1.309 ≈ 15.708 m (exactly 5π m).
  3. Sector area: A = ½ × 12² × 1.309 ≈ 94.2478 m².
  4. Perimeter of the slice: P = 2 × 12 + 15.708 ≈ 39.708 m, and the straight-line chord is c = 24 × sin 37.5° ≈ 14.6103 m.

Result: The path is about 15.708 m long and the sector of lawn covers about 94.2478 m².

Understanding the result

The arc length is measured along the curve, so it is always longer than the chord between the same two points; the gap grows as the angle widens. For small angles the two are almost equal, which is why short arcs on large circles look nearly straight.

The share of the full circle tells you what fraction of the circumference and of the area the sector takes: a 90° sector is 25% of both.

Assumptions and limitations

  • The arc belongs to a perfect circle with the radius you enter.
  • In chord mode the minor arc (the shorter of the two possible arcs) is used.
  • Results are rounded only for display.
  • Angles above one full turn are rejected, since a sector cannot overlap itself.
  • Arcs of other curves, such as ellipses or parabolas, need calculus and are not covered.
  • There is no built-in length-unit conversion; convert values first if needed.

Frequently asked questions

Why must the angle be in radians for s = rθ?

Radians measure an angle by the arc it cuts off in units of the radius, so the formula is really a definition. If you use degrees, you have to include the conversion factor: s = r × θ° × π ÷ 180.

What is the difference between arc length and chord length?

The arc is the curved distance along the circle; the chord is the straight line between the same two endpoints. For a 60° arc on a circle of radius 10, the arc is about 10.47 while the chord is exactly 10.

How do I find the arc length if I only know the diameter?

Halve the diameter to get the radius and enter that. Every formula here uses the radius.

Can I get the angle from the arc length?

Yes: rearrange s = rθ to θ = s ÷ r, which gives radians. Multiply by 180/π for degrees. The calculator's solve-for option can also find the angle that produces a target arc length.

How this calculator is checked

Method: Radian definition s = rθ; sector area ½r²θ; chord 2r sin(θ/2). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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