Exact length of a specified chord of a circular curve For a circular curve of radius R and degree of curve D (angle subtended at the center), the exact length of the specified chord is:

Introduction / Context:
Chord lengths are frequently required in curve setting from the central angle. The exact relationship with the radius and subtended angle is a basic trigonometric identity used in field computations and setting-out sheets.

Given Data / Assumptions:

• Simple circular geometry.
• Central angle equals the degree of the specified chord (in degrees or radians).
• Standard chord formula applies without small-angle approximations.

Concept / Approach:

The chord subtending a central angle D has length c = 2R * sin(D/2). Since the diameter is 2R, an equivalent statement is c = diameter × sin(D/2). This is exact, not approximate.

Step-by-Step Solution:

Verification / Alternative check:

For small D, sin(D/2) ≈ D/2 (radians), giving c ≈ R * D, aligning with arc approximation c ≈ s for small angles.

Why Other Options Are Wrong:

R × sin(D/2) misses the factor 2.

Diameter × cos(D/2) and Diameter × tan(D/2) are unrelated to chord length.

“None of these” is invalid because the correct identity is known.

Common Pitfalls:

Confusing chord with arc length; mixing degrees and radians when evaluating numerically.

Final Answer:

Diameter × sin(D/2)