Area of a Parabolic Segment — Formula & Examples

Area of a Parabolic Segment

The formula is given below.

A parabolic segment: parabolic arc bounded below by a chord (horizontal line). Width labelled along the chord, height labelled vertically from chord to arc apex. Area = ⅔ × width × height.

See also

Key Formula

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

Where:

$A$ = Area of the parabolic segment
$b$ = Length of the chord (base) that cuts across the parabola
$h$ = Height of the segment, measured as the perpendicular distance from the chord to the point on the parabola farthest from the chord

Worked Example

Problem: A parabolic arch has a base (chord) of 6 meters and a height of 4 meters. Find the area of the parabolic segment.

Step 1: Identify the base and height of the parabolic segment.

b = 6, \quad h = 4

Step 2: Apply the parabolic segment area formula.

A = \frac{2}{3} \, b \, h = \frac{2}{3} \times 6 \times 4

Step 3: Calculate the result.

A = \frac{48}{3} = 16

Answer: The area of the parabolic segment is 16 square meters.

Why It Matters

Archimedes' result for the parabolic segment was one of the earliest examples of finding a curved area using rigorous methods, centuries before calculus was invented. Engineers and architects use this formula when designing parabolic arches, bridges, and reflectors, where knowing the enclosed area is essential for structural and material calculations.

Common Mistakes

Mistake: Using 1/3 instead of 2/3 in the formula.

Correction: The area of a parabolic segment is 2/3 of the enclosing rectangle (b × h), not 1/3. The fraction 1/3 applies to a triangle's relationship with a rectangle, not a parabolic segment.

Related Terms

Parabola — The curve that defines the segment's boundary
Formula — General term for mathematical relationships
Area — The quantity being computed
Chord — The straight line forming the segment's base
Integration — Calculus method that can also derive this area