Length of an arc of a parabola

If a point X is located on a parabola which has focal length and if is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at X can be calculated from and as follows, assuming they are all expressed in the same units.

This quantity, , is the length of the arc between X and the vertex of the parabola.
The length of the arc between X and the symmetrically opposite point on the other side of the parabola is
The perpendicular distance, , can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign of reverses the signs of and without changing their absolute values. If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of

This can be useful, for example, in calculating the size of the material needed to make a parabolic reflector or parabolic trough.

(Note: In the above calculation, the square-root, , must be positive. The quantity ln(a), sometimes written as log_e(a), is the natural logarithm of a, i.e. its logarithm to base "e".)