Area 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 area $\AA$ of $ABC$ is given by:

$\AA = \dfrac {2 a b} 3$

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

Construct the triangle $\triangle ABC$:

From Quadrature of Parabola:

$\AA = \dfrac 4 3 \triangle ABC$

From Area of Triangle in Terms of Side and Altitude, the area of $\triangle ABC$ equals $\dfrac {a b} 2$.

The result follows.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Segment of a Parabola: $4.24$
• 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: $7.24.$
• Weisstein, Eric W. "Parabolic Segment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParabolicSegment.html