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The formula for long chord length can be given as __________

$L = 2R \times \tan \frac{∆}{2}$

$L = 2 R \times \sin \frac{∆}{2}$

$L = R ( s e c \frac{∆}{2} - 1 )$

$L = R ( 1 - \cos \frac{∆}{2} )$

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilAdvanced Survey

Option B

$L = 2 R \times \sin \frac{∆}{2}$

Quick Summary:

The long chord ( $L$ ) of a simple circular curve is the straight line distance joining the point of curve (PC) to the point of tangency (PT) · It represents the longest chord formed between the initial and final tangent points of a simple circular curve · For a curve of radius $R$ and deflection angle $\Delta$ , the formula for long chord length is given by $L = 2R \sin\left(\frac{\Delta}{2}\right)$ .

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$L = 2R \sin\left(\frac{\Delta}{2}\right)$ — Long chord length

$T = R \tan\left(\frac{\Delta}{2}\right)$ — Tangent length

$l = \frac{\pi R \Delta}{180°$ — Length of curve

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

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

⚙️ Working Principle

Consider the isosceles triangle formed by the center of the curve $O$ , the Point of Curve $T_1$ , and the Point of Tangency $T_2$ . The angle subtended at the center by the long chord $T_1T_2$ is equal to the deflection angle $\Delta$ . Bisecting this central angle creates two congruent right-angled triangles with hypotenuse equal to radius $R$ and opposite side equal to half the long chord length ( $L/2$ ) · Applying simple trigonometry, $\sin\left(\frac{\Delta}{2}\right) = \frac{L/2}{R}$ , which yields $L = 2R \sin\left(\frac{\Delta}{2}\right)$ .

📌 Key Points

▸
Long chord joins the initial point of curve (PC or $T_1$ ) and the point of tangency (PT or $T_2$ ).
▸
It bisects the central angle subtended by the curve into two equal angles of $\Delta/2$ .
▸
It forms the baseline when setting out simple circular curves using the method of offsets from the long chord.

🛠️ Applications / Uses

▸
Design and layout of horizontal simple circular curves in roads and railways.
▸
Setting out curves using the offset from long chord method.

📄 Additional Information

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Option A ( $L = 2R \tan\left(\frac{\Delta}{2}\right)$ ) is incorrect as $R \tan\left(\frac{\Delta}{2}\right)$ represents the single Tangent Length ( $T$ ).
▸
Option C ( $L = R \left(\sec\left(\frac{\Delta}{2}\right) - 1\right)$ ) represents the External or Apex Distance ( $E$ ).
▸
Option D ( $L = R \left(1 - \cos\left(\frac{\Delta}{2}\right)\right)$ ) represents the Mid-Ordinate ( $M$ ).

📊 Diagram / Illustration

✅

B is correct — The length of the long chord is given by $L = 2R \sin\left(\frac{\Delta}{2}\right)$ , derived from right-triangle trigonometry on the bisected central deflection angle.

Core Concepts Used

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Simple Circular Curve Geometry Deflection Angle and Central Angle Relationship Elements of Curve Surveying

💡 EXAM TIP

In competitive exam questions, carefully distinguish between Mid-Ordinate ( $M = R(1 - \cos(\Delta/2))$ ), Apex Distance ( $E = R(\sec(\Delta/2) - 1)$ ), Tangent Length ( $T = R \tan(\Delta/2)$ ), and Long Chord ( $L = 2R \sin(\Delta/2)$ ) as options frequently swap these trigonometric functions.

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