Parabolic Segment — Definition, Formula & Examples

A parabolic segment is the region enclosed between a parabola and a straight line (called a chord) that intersects the parabola at two points.

Given a parabola and a chord connecting two points on it, the parabolic segment is the bounded planar region whose boundary consists of the chord and the parabolic arc between those two points. Its area equals two-thirds the area of the circumscribed parallelogram (or equivalently, two-thirds the base times the height of the segment), a result first proved by Archimedes.

Key Formula

A = \frac{2}{3}\,b\,h

Where:

$A$ = Area of the parabolic segment
$b$ = Length of the chord (base of the segment)
$h$ = Perpendicular distance from the chord to the vertex of the parabolic arc

How It Works

To find the area of a parabolic segment, you need the length of the chord (the base

b

) and the perpendicular distance from the chord to the vertex of the parabolic arc (the height

h

). The area is then

\frac{2}{3}bh

. This remarkably simple formula holds regardless of the orientation of the parabola or the angle of the chord, as long as

b

and

h

are measured correctly.

Worked Example

Problem: Find the area of the parabolic segment cut from the parabola y = x² by the horizontal line y = 9.

Find the endpoints: Set x² = 9 to find where the chord meets the parabola.

x = -3 \quad \text{and} \quad x = 3

Determine base and height: The chord runs from (−3, 9) to (3, 9), so b = 6. The vertex of the parabolic arc is at (0, 0), and its perpendicular distance to the chord y = 9 is h = 9.

b = 6, \quad h = 9

Apply the formula: Substitute into the area formula.

A = \frac{2}{3}(6)(9) = 36

Answer: The area of the parabolic segment is 36 square units. (You can verify this by integration: ∫₋₃³ (9 − x²) dx = 36.)

Why It Matters

Archimedes' parabolic segment formula is one of the earliest known results equivalent to integration, making it a milestone in the history of calculus. In engineering and architecture, parabolic segments appear in bridge arches, satellite dishes, and reflector designs where computing enclosed areas is essential.

Common Mistakes

Mistake: Using ½bh (the triangle formula) instead of ⅔bh for the parabolic segment.

Correction: A parabolic segment always has area exactly ⅔bh — it fills two-thirds of the bounding rectangle, not one-half. The ½bh formula applies to triangles, not parabolic regions.