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Calculate the radius for a curve having long chord 50 m and mid ordinate 3m.

$105.67\text{ m}$

$106.67\text{ m}$

$107.67\text{ m}$

$108.67\text{ m}$

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilAdvanced Survey

Option A

$105.67\text{ m}$

Quick Summary:

The correct answer is Option A (105.67 m). The radius of a simple circular curve can be determined directly from the geometric relationship between the length of the long chord ( $L$ ) and the mid-ordinate ( $M$ ). Substituting the given values $L = 50\text{ m}$ and $M = 3\text{ m}$ into the exact geometric formula gives $R = 105.67\text{ m}$ .

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💡 Explanation

🔢 Key Formulas

$R = \frac{M}{2} + \frac{L^2}{8M}$ — Exact relation between radius ( $R$ ), long chord ( $L$ ), and mid-ordinate ( $M$ )

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

$L = 2R \sin\frac{\Delta}{2}$ — Long chord in terms of radius and deflection angle

⚙️ Working Principle

In a circular arc, the mid-ordinate represents the perpendicular distance from the midpoint of the long chord to the apex of the curve. By applying the Pythagorean theorem to the right-angled triangle formed by the center of the curve, half the long chord, and the radius minus the mid-ordinate, a direct quadratic relationship between radius, chord length, and mid-ordinate is established.

📌 Key Points

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The exact geometric expression for radius is derived from $R^2 = (R - M)^2 + (L/2)^2$ .
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Expanding the relation yields $R^2 = R^2 - 2RM + M^2 + \frac{L^2}{4}$ , which simplifies to $2RM = M^2 + \frac{L^2}{4}$ .
▸
Dividing both sides by $2M$ yields $R = \frac{M}{2} + \frac{L^2}{8M}$ .
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An approximate formula often used in field surveying when $M$ is very small is $R \approx \frac{L^2}{8M}$ .

✅ Advantages

▸
Allows direct calculation of radius without requiring deflection angle measurements.
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Useful for setting out circular curves using the offset from long chord method in fieldwork.

❌ Disadvantages / Limitations

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Requires high precision in measuring mid-ordinate $M$ as small errors significantly affect calculated radius.
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Method becomes cumbersome for very long curves with large offsets.

🛠️ Applications / Uses

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Determining circular curve radius in highway and railway route alignment.
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Setting out curves using offsets from the long chord in field surveying.

📄 Additional Information

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Using approximate formula $R \approx \frac{L^2}{8M} = \frac{50^2}{8 \times 3} = 104.17\text{ m}$ gives an error due to ignoring the $\frac{M}{2}$ term.
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Option B (106.67 m), Option C (107.67 m), and Option D (108.67 m) are incorrect values resulting from arithmetic miscalculations or applying incorrect approximations.

📊 Diagram / Illustration

✅

A is correct — Using $R = \frac{M}{2} + \frac{L^2}{8M}$ with $L = 50\text{ m}$ and $M = 3\text{ m}$ , $R = 1.5 + \frac{2500}{24} = 105.67\text{ m}$ .

Core Concepts Used

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Elements of Simple Circular Curve Mid-ordinate and Long Chord Relationships Setting Out Curves by Offsets from Long Chord

💡 EXAM TIP

Always check if the question requires the exact formula $R = \frac{M}{2} + \frac{L^2}{8M}$ or the approximate formula $R \approx \frac{L^2}{8M}$ ; options with fine decimal differences like 105.67 m vs 104.17 m specifically test the exact formula.

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