Arc Length of Right Parabolic Segment

Theorem

Let $ABC$ be a right parabolic segment where:

$AB$ is the defining chord $\LL$ of $ABC$

$C$ is the vertex of the defining parabola $\PP$ of $ABC$.

The length $\LL$ of the arc $ACB$ is given by:

$\ds \LL$

$=$

$\ds \dfrac {\sqrt {b^2 + 16 a^2} } 2 + \dfrac {b^2} {8 a} \map \ln {\dfrac {4 a + \sqrt {b^2 + 16 a^2} } b}$

$\ds $

$=$

$\ds \dfrac {\sqrt {b^2 + 16 a^2} } 2 + \dfrac {b^2} {2 a} \inv \sinh {\dfrac {2 a} b}$

where:

$a$ is the length of the line segment $CF$, where $F$ is the point at which the axis of $\PP$ intersects $AB$

$b$ is the length of the line segment $AB$.

Proof

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Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Segment of a Parabola: $4.25$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Segment of a Parabola: $7.25.$
Weisstein, Eric W. "Parabolic Segment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParabolicSegment.html