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The formula for mid ordinate length can be given as __________

$M = 2R \times \tan \frac{∆}{2}$

$M = 2 R \times \sin \frac{∆}{2}$

$M = R ( s e c \frac{∆}{2} - 1 )$

$M = R ( 1 - \cos \frac{∆}{2} )$

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilAdvanced Survey

Option D

$M = R ( 1 - \cos \frac{∆}{2} )$

Quick Summary:

The mid-ordinate ( $M$ ) of a simple circular curve is the ordinate from the midpoint of the long chord to the midpoint of the curve · It is also known as the versed sine of the curve · The correct expression for calculating the mid-ordinate length is $M = R\left(1 - \cos\frac{\Delta}{2}\right)$ .

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$M = R\left(1 - \cos\frac{\Delta}{2}\right)$ — Length of Mid-Ordinate

$T = R \tan\frac{\Delta}{2}$ — Tangent Length

$E = R\left(\sec\frac{\Delta}{2} - 1\right)$ — Apex Distance (External Distance)

$L = R \times \Delta \times \frac{\pi}{180°$ — Length of Curve

$L_c = 2R \sin\frac{\Delta}{2}$ — Length of Long Chord

⚙️ Working Principle

In a simple circular curve, the distance from the center of the curve $O$ to the midpoint of the curve is the radius $R$ . The distance from the center $O$ to the midpoint of the long chord is $R\cos\left(\frac{\Delta}{2}\right)$ . Subtracting this from the radius gives $M = R - R\cos\left(\frac{\Delta}{2}\right) = R\left(1 - \cos\frac{\Delta}{2}\right)$ .

📌 Key Points

▸
The mid-ordinate is the perpendicular distance between the midpoint of the long chord and the apex of the curve.
▸
Mid-ordinate $M$ is also called the versed sine of half the deflection angle.

🛠️ Applications / Uses

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Setting out simple circular curves in field surveying using the offset from long chord method.
▸
Determining sight distance requirements on horizontal curves in highway engineering.

📄 Additional Information

▸
Option A ( $2R\tan\frac{\Delta}{2}$ ) is incorrect; tangent length is $R\tan\frac{\Delta}{2}$ .
▸
Option B ( $2R\sin\frac{\Delta}{2}$ ) represents the total length of the long chord ( $L_c$ ).
▸
Option C ( $R(\sec\frac{\Delta}{2}-1)$ ) represents the apex distance or external distance ( $E$ ).

📊 Diagram / Illustration

✅

D is correct — The formula for mid-ordinate length is $M = R\left(1 - \cos\frac{\Delta}{2}\right)$ .

Core Concepts Used

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Simple Circular Curve Geometry Mid-Ordinate (Versed Sine) Curve Elements in Surveying

💡 EXAM TIP

Remember that external distance $E$ uses $(\sec\frac{\Delta}{2} - 1)$ , while mid-ordinate $M$ uses $(1 - \cos\frac{\Delta}{2})$ . Since $\cos$ and $\sec$ are reciprocals, note that $E = R\left(1 + \frac{M}{R\cos(\Delta/2)} - 1\right)$ relates both parameters geometrically.

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