Arc Length Calculator
Find the length of a circular arc and the area of its sector from a radius and a central angle, or recover the angle from a known arc or sector. In radians, L = rθ and A = (1/2)r²θ. In degrees, take the fraction θ/360 of the full circumference and the full disk.
Sector A = 13.08996939 · θ = 60°
- Convert the central angle to radians: θ = 60 × π / 180 = 1.0471975512.
- Arc length L = rθ = 5 × 1.0471975512 = 5.23598775598. That is also (60/360) × 2πr.
- Sector area A = (1/2)r²θ = 13.08996939. That is also (60/360) × πr².
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How to Use the Arc Length Calculator
From r and an angle (degrees or radians) you get the arc and the sector. From r and the arc, or r and the sector, you get the central angle.
- Choose the pair you know.
- Enter a non-negative radius. Angles stay between 0 and one full turn (360° or 2π).
- Read the arc (or the angle), the sector area, and the chord that closes the slice.
The full circumference and the full disk belong on the Circle Calculator (C = 2πr, A = πr²). Converting the square units after you have a sector belongs on the Area Converter. A triangle’s area belongs on the Triangle Area Calculator.
Arc and Sector
A sector is a pizza slice: two radii and the arc between them. The arc is the crust of that slice. Both are the same fraction of the whole circle as the central angle is of a full turn. One-sixth of 360° is 60°, so the 60° sector is one-sixth of πr² and the 60° arc is one-sixth of 2πr.
The radian formulas are shorter because a radian is defined so that the arc equals rθ. Solving for the angle is a division: θ = L / r, or θ = 2A / r². The chord that closes the slice is 2r sin(θ/2); at 60° that chord equals the radius.
Arc Length Calculator Examples
Radius 5, 60°
L = 5π/3 ≈ 5.23598775598. Sector A = 25π/6 ≈ 13.08996939. Chord = 5.
Radius 5, 90° (a quarter)
L = 5π/2 ≈ 7.85398163397. Sector A = 25π/4 ≈ 19.6349540849. One-fourth of the r = 5 circle on the Circle Calculator.
Radius 5, 180° (a semicircle)
L = 5π ≈ 15.7079632679. Sector A = 25π/2 ≈ 39.2699081699. The chord is the diameter, 10.
Radius 5, π/3 radians
π/3 rad is 60°. Same arc and sector as the first example. The radian box does not convert again.
Arc of the 60° slice, known radius
Entering L ≈ 5.23598775598 and r = 5 returns θ = 60°.
A negative angle
Direction around the circle is out of scope. The calculator rejects −10 instead of using |θ|.
Degrees vs Radians
Classroom geometry usually gives the angle in degrees. Then the safe move is the fraction of the circle: θ/360. Calculus and the radian formula L = rθ expect θ already in radians. Feeding 60 into L = rθ treats the angle as 60 radians — almost ten full turns — and the arc comes out about 60 times too long.
This page converts degrees to radians as θ × π / 180 before it multiplies by r. 180° is π radians. 360° is 2π, the full circumference on the Circle Calculator. A major arc (the long way around) is just a large central angle: 300° is the complement of a 60° minor arc on the same circle.
Common Arc and Sector Mistakes
Using degrees inside L = rθ
Convert first, or use (θ/360) × 2πr. L = 5 × 60 = 300 is not the 60° arc.
Using the full circumference as the arc
C = 2πr is the whole rim, on the Circle Calculator. A 60° arc is one-sixth of that.
Reporting the sector as πr²
That is the full disk. Multiply by θ/360, or use (1/2)r²θ with θ in radians.
Confusing arc length with chord length
The arc follows the rim. The chord is the straight cut. At 60° and r = 5 they are ≈ 5.236 and 5 — close, not equal except in the small-angle limit.
An angle larger than 360°
One full turn is the whole circle. This page stops at 360° / 2π.
Arc Length Calculator FAQs
What is the formula for arc length?
In radians, L = rθ. In degrees, L = (θ/360) × 2πr. For r = 5 and 60°, L = 5π/3 ≈ 5.23598775598.
What is the area of a sector?
In radians, A = (1/2)r²θ. In degrees, A = (θ/360) × πr². For r = 5 and 60°, A = 25π/6 ≈ 13.08996939.
How do you find the central angle from the arc?
θ = L / r in radians, then multiply by 180/π for degrees.
Is a 90° sector one-fourth of the circle?
Yes. 90/360 = 1/4, so the arc is C/4 and the sector is A/4 of the same radius.
Is this the same as the Circle Calculator?
No. Circle is the full rim and the full disk. This page is a slice: one arc and its sector.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.