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What would be the value of versed distance if the long chord is given as 80m and the radius of the curve is given as 200m?
041 m
041 m
041 m
041 m
041 m
The versed distance (or versed sine / mid-ordinate) of a circular curve is the perpendicular distance from the midpoint of the long chord to the apex of the curve · It is calculated using the relation O0=R−R2−(L/2)2, where R is the radius of the curve and L is the length of the long chord · For a chord of 80 m and radius of 200 m, the versed distance evaluates to $4.041\text{ m} (commonly rounded/represented as \4.041\text{ m}or041\text{ m}$ format in exam options).
The versed distance (or versed sine / mid-ordinate) of a circular curve is the perpendicular distance from the midpoint of the long chord to the apex of the curve · It is calculated using the relation O0=R−R2−(L/2)2, where R is the radius of the curve and L is the length of the long chord · For a chord of 80 m and radius of 200 m, the versed distance evaluates to $4.041\text{ m} (commonly rounded/represented as \4.041\text{ m}or041\text{ m}$ format in exam options).
O0=R−R2−(2L)2 — Exact formula for Versed Distance (Mid-Ordinate)
O0≈8RL2 — Approximate formula for Versed Distance when L≪R
In a right-angled triangle formed by the center of the curve, the midpoint of the chord, and one endpoint of the chord, the base is half the chord length (L/2), the hypotenuse is the radius (R), and the adjacent side along the central line is R2−(L/2)2. Subtracting this adjacent distance from the total radius (R) gives the mid-ordinate or versed distance (O0).
Versed sine (sagitta) is the distance along the radial line from the midpoint of the long chord to the curve apex.
Using exact formula: O0=200−2002−402=200−38400≈4.041 m.
Using approximate formula: O0≈8×200802=16006400=4 m.
Allows simple setting out of circular curves using ordinates from the long chord without needing a theodolite.
Useful for short-radius curves and minor road/rail alignments in field surveys.
Inaccurate for very long curves where offsets become excessively large.
Requires open, unobstructed line of sight along the entire long chord.
Ranging circular curves in highway and railway alignment.
Setting out curve points using offset measurements along the long chord.
The exact value is $4.041\text{ m}$. In exam questions, options are often abbreviated or stylized as 4.041 m or formatted as 041 m due to OCR/print truncation.
Approximate formula O0≈8RL2 yields 4 m, while the exact value is $4.041\text{ m}$.
A is correct — Using O0=R−R2−(L/2)2, O0=200−2002−402≈4.041 m.
Remember that O0≈8RL2 is a quick approximation for exam numericals, but always use the exact formula O0=R−R2−(L/2)2 when precision to decimal places is required.