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Horizontal Curve Middle Ordinate

M=R(1−cos⁡Δ2)M = R\left(1 - \cos\frac{\Delta}{2}\right)
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Worked example: 50 m ordinate on a 100 m radius → Δ = 120°. Press Try an example to run it live, then adjust anything.

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See also

Horizontal Curve External Distance: Both use curve radius and deflection angle.Horizontal Curve Long Chord: Both use curve radius and deflection angle.Horizontal Curve Tangent Length: Both use curve radius and deflection angle.Horizontal Curve Length from Degree of Curve: Both use deflection angle.Deflection Angle to a Point on a Circular Curve: Both use curve radius.Minimum Radius of a Horizontal Curve at a Design Speed (AASHTO): Both use curve radius.
2 more relatedRadius from Degree of Curve (Arc Definition): Both use curve radius.Superelevation Rate for a Horizontal Curve: Both use curve radius.
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Horizontal Curve Middle Ordinate explained

ΔRM

The middle ordinate is the sagitta, the "arrow" of the arc, and it is the quantity that decides whether a driver can see around the inside of a curve. Set M equal to the horizontal clearance from the centreline to a sight obstruction (a cut slope, a noise wall, a row of trees) and the same relation tells you the length of curve over which the sightline is blocked. Highway design manuals print it as a nomograph of R against M for exactly this purpose.

A worked example: a 500 ft curve deflecting 60° has M = 500(1 − cos 30°) = 66.99 ft, so the arc bulges 67 ft off its own long chord. Field crews use the same idea in reverse to measure a curve that was never staked: stretch a chord across a rail or a curb, measure the offset at midspan, and R follows from R ≈ C²/(8M), the small-angle version of this formula. That approximation, incidentally, is how track inspectors measure curvature with a string and a ruler.

Horizontal Curve Middle Ordinate formula

M=R(1−cos⁡Δ2)M = R\left(1 - \cos\frac{\Delta}{2}\right)
Where
  • MM= Middle ordinate (m)
  • RR= Curve radius (m)
  • Δ\Delta= Deflection angle (°)
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Horizontal curve layout

6 formulasSetting out a circular road curve: radius from degree of curve, then tangent, curve length, external distance, middle ordinate and long chord.

Revised 16 October 2026
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formula.expert. (2026, October 16). Horizontal Curve Middle Ordinate Formula & Calculator. https://formula.expert/solvers/circular-curve-middle-ordinate

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