The formula for long chord length can be given as __________ | Civil MCQ | Examoogle

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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 $L = 2R \sin\left(\frac{\Delta}{2}\right)$ .

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💡 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

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

🛠️ Applications / Uses

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Design and layout of horizontal simple circular curves in roads and railways.
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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 ( $T$ ).
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Option C ( $L = R \left(\sec\left(\frac{\Delta}{2}\right) - 1\right)$ ) represents the External or Apex Distance ( $E$ ).
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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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