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The formula for tangent length can be given as __________

$T = R \times \tan \frac{∆}{2}$

$T = 2 R \times \sin \frac{∆}{2}$

$T = R ( s e c \frac{∆}{2} - 1 )$

$T = R ( 1 - \cos \frac{∆}{2} )$

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilAdvanced Survey

Option A

$T = R \times \tan \frac{∆}{2}$

Quick Summary:

In simple circular curve surveying, the tangent length ( $T$ ) represents the distance from either point of curvature (PC or PT) to the point of intersection (PI) · It is determined by the radius of the curve ( $R$ ) and the deflection angle (or intersection angle, $\Delta$ ).

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

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

$L = \frac{\pi R \Delta}{180°$ — Length of the curve ( $L$ )

$C = 2R \sin\left(\frac{\Delta}{2}\right)$ — Length of long chord ( $C$ )

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

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

⚙️ Working Principle

From the geometric properties of a simple circular curve, the intersection lines form a right-angled triangle with the radius drawn to the tangent points · The tangent length is opposite to half the deflection angle ( $\Delta/2$ ) relative to the center, leading to the trigonometric relationship $T = R \tan(\Delta/2)$ .

📌 Key Points

▸
The tangent points are designated as T₁ (Point of Curve) and T₂ (Point of Tangency).
▸
The tangent distance is measured along the straight line from the Point of Intersection (PI) to either tangent point.

🛠️ Applications / Uses

▸
Setting out simple circular curves in railway and highway alignment designs.
▸
Determining the chainage of starting and ending curve points during field surveying.

📄 Additional Information

▸
Option B ( $2R \sin(\Delta/2)$ ) is the formula for the Length of the Long Chord ( $C$ ).
▸
Option C ( $R(\sec(\Delta/2) - 1)$ ) is the formula for Apex Distance or External Distance ( $E$ ).
▸
Option D ( $R(1 - \cos(\Delta/2))$ ) is the formula for Mid-Ordinate distance ( $M$ ).

📊 Diagram / Illustration

✅

A is correct — The tangent length of a simple circular curve is directly calculated as $T = R \tan(\Delta/2)$ .

Core Concepts Used

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Simple Circular Curves Elements of Curve Deflection Angle and Tangent Distance

💡 EXAM TIP

Memorize the core trigonometrical multipliers for curve elements relative to $\Delta/2$ : $\tan(\Delta/2)$ for Tangent Length, $\sin(\Delta/2)$ for Chord/2, $\sec(\Delta/2)-1$ for Apex Distance, and $1-\cos(\Delta/2)$ for Mid-Ordinate.

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