Learn: Circular-curve long chord
The long chord is the straight line between the start and end of a circular curve. It is useful for geometric checks but does not equal distance traveled along the arc.
What the formula is saying
Bisect the central angle to form two right triangles. Each half-chord equals R sin(Δ/2), so doubling gives the full chord.
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.
- LC
- Result to find
Circular-curve long chord. Scale the dimensionless geometric multiplier by the radius with the stated numerical factor.
m
Sort out the units first
R and the requested long chord 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 LC and explain the result in the stated output unit.
- R · Curve radius
- 300 m
- Δ · Central angle
- 40 deg
Convert half the central angle to radians
The symmetric curve geometry uses half the full central or deflection angle.
(40) × π ÷ 360 ≈ 0.3490658504 radEvaluate the geometric multiplier
Apply the trigonometric function to the radian half-angle before multiplying by radius.
sin((0.3490658504)) ≈ 0.3420201433Calculate the long chord
Scale the dimensionless geometric multiplier by the radius with the stated numerical factor.
2 × (300) × (0.3420201433) ≈ 205.212086 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; check that the result refers to the geometric feature requested, not another curve element.
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 half the central angle to radians
The symmetric curve geometry uses half the full central or deflection angle.
(60) × π ÷ 360 ≈ 0.5235987756 radEvaluate the geometric multiplier
Apply the trigonometric function to the radian half-angle before multiplying by radius.
sin((0.5235987756)) = 0.5Calculate the long chord
Scale the dimensionless geometric multiplier by the radius with the stated numerical factor.
2 × (250) × (0.5) = 250 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 long chord
A hint, not the answer
Bisect the central angle to form two right triangles. Each half-chord equals R sin(Δ/2), so doubling gives the full chord.
R and the requested long chord 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 half the central angle to radians
The symmetric curve geometry uses half the full central or deflection angle.
(30) × π ÷ 360 ≈ 0.2617993878 radEvaluate the geometric multiplier
Apply the trigonometric function to the radian half-angle before multiplying by radius.
sin((0.2617993878)) ≈ 0.2588190451Calculate the long chord
Scale the dimensionless geometric multiplier by the radius with the stated numerical factor.
2 × (400) × (0.2588190451) ≈ 207.0552361 m
Avoid the common trap
Do not omit the factor 2 or use sin Δ instead of sin(Δ/2). A chainage distance along the curve is an arc, not this straight chord.
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.
This optional checkmark is saved only in this browser. It is your own progress note, not a certificate.
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 long chord. 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.