Learn: Circular-curve arc length
Arc length measures distance along the curved alignment between the beginning and end of a circular curve. It is longer than the straight chord joining those endpoints.
What the formula is saying
A full circumference 2πR corresponds to 360°. Take the fraction Δ/360 of that circumference, which simplifies to πRΔ/180.
Read the symbols in plain language
- R
- Curve radius
Horizontal radius from the circle centre to the alignment arc; keep the same length unit throughout the circular-curve calculation.
mMetres measure length; 1 m = 1000 mm.
- Δ
- Central angle
Positive angular sweep of the simple circular curve in degrees; this lesson uses the shorter arc with an angle strictly below 180°.
degAngles are entered in degrees; multiply by π/180 for trigonometric calculations in radians.
- L
- Result to find
Circular-curve arc length. Multiplying radius by the radian angle gives the curved length rather than its chord.
m
Sort out the units first
R and the requested arc length use m. Δ is the full central/deflection angle entered in degrees, represented by input D. It is an angle, not a pipe diameter or degree-of-curve parameter.
Assumptions before calculating
Assume one simple horizontal circular curve with positive radius R and a deflection or central angle Δ strictly between 0° and 180°. Transition spirals, compound curves and vertical geometry are not included.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find L and explain the result in the stated output unit.
- R · Curve radius
- 300 m
- Δ · Central angle
- 40 deg
Convert the central angle to radians
Arc length equals radius times central angle only when the angle is in radians.
(40) × π ÷ 180 ≈ 0.6981317008 radCalculate the distance along the arc
Multiplying radius by the radian angle gives the curved length rather than its chord.
(300) × (0.6981317008) ≈ 209.4395102 m
Does this worked answer make sense?
At the same angle, doubling R doubles the requested distance. For a small positive angle the distance remains positive; the arc must exceed its corresponding long chord.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- R · Curve radius
- 250 m
- Δ · Central angle
- 60 deg
Convert the central angle to radians
Arc length equals radius times central angle only when the angle is in radians.
(60) × π ÷ 180 ≈ 1.047197551 radCalculate the distance along the arc
Multiplying radius by the radian angle gives the curved length rather than its chord.
(250) × (1.047197551) ≈ 261.7993878 m
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- R · Curve radius
- 400 m
- Δ · Central angle
- 30 deg
Find: Learn: Circular-curve arc length
A hint, not the answer
A full circumference 2πR corresponds to 360°. Take the fraction Δ/360 of that circumference, which simplifies to πRΔ/180.
R and the requested arc length use m. Δ is the full central/deflection angle entered in degrees, represented by input D. It is an angle, not a pipe diameter or degree-of-curve parameter.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Convert the central angle to radians
Arc length equals radius times central angle only when the angle is in radians.
(30) × π ÷ 180 ≈ 0.5235987756 radCalculate the distance along the arc
Multiplying radius by the radian angle gives the curved length rather than its chord.
(400) × (0.5235987756) ≈ 209.4395102 m
Avoid the common trap
Do not use the chord formula for chainage along the curve. A degree value must be converted; multiplying R by Δ directly only works if Δ is already in radians.
When this method applies — and when it does not
This is exact plane-circle geometry for the specified curve, not a complete road alignment design. Tangent points, survey datum, transitions, sight distance, superelevation and applicable design criteria must be established separately.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
One idea understood. Keep going.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Circular-curve arc length. Horizontal/vertical alignment and sight-distance context. The geometry identities here are not a selection of a compliant speed, radius or K value for a real road.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.