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What would be the value of apex distance if the angle is given as 55° and the radius of the curve is given as 20m?
23.318 m
547 m
14.904 m
29.808 m
23.318 m
The apex distance (or external distance) of a circular curve is the distance from the point of intersection (PI) to the midpoint of the curve (apex). When given an intersection angle I=55°, the deflection angle is Δ=180°−55°=125°. Substituting R=20 m and Δ=125° into the apex distance formula gives E=23.318 m.
The apex distance (or external distance) of a circular curve is the distance from the point of intersection (PI) to the midpoint of the curve (apex). When given an intersection angle I=55°, the deflection angle is Δ=180°−55°=125°. Substituting R=20 m and Δ=125° into the apex distance formula gives E=23.318 m.
E = R \left(\sec\left( rac{\Delta}{2} ight) - 1\right) — Apex distance (External distance)
Δ=180°−I — Relation between deflection angle (Δ) and intersection angle (I)
In highway curve geometry, the apex distance E represents the radial clearance between the point of intersection of two straight tangents and the summit of the circular arc. It is determined using trigonometric relationships in the right-angled triangle formed by the center of the curve, the midpoint of the curve, and the point of intersection.
Apex distance is also called external distance in circular curve alignment.
If the given angle is the intersection angle I, the deflection angle Δ equals 180°−I.
It measures the vertical distance from the vertex (PI) to the curve apex.
Designing horizontal highway and railway alignment curve clearances.
Determining sight distances and buffer space at intersection turns.
If the 55° angle was taken directly as the deflection angle Δ, the value would be 20imes(sec(27.5°)−1)=2.548 m.
Option B (547 m), C (14.904 m), and D (29.808 m) represent incorrect trigonometric calculations or misidentifications of curve elements like tangent length or long chord.
A is correct — Using deflection angle Δ=180°−55°=125°, the apex distance formula yields E=20imes(sec(62.5°)−1)=23.318 m.
Always verify whether the angle provided in survey problems is the deflection angle (Δ) or the interior angle of intersection (I=180°−Δ) before applying trigonometric formulas.
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