Geometry of a partially full circular sewer: If the central angle of the water-filled segment is α (in radians) for a pipe of radius r, what is the cross-sectional flow area A?

Introduction / Context:
Many hydraulic calculations for sewers flowing partially full require geometric properties of a circular segment: area, wetted perimeter, and hydraulic radius as functions of the central angle α. These relationships enable evaluation of conveyance using Manning or Chézy formulas.

Given Data / Assumptions:

• Circular pipe of radius r.
• Water-filled segment subtends angle α at the center (radians).
• Uniform cross-section; ignore crown roughness variations.

Concept / Approach:

The area of a circular segment equals the area of the sector minus the area of the isosceles triangle formed by the two radii and the chord. Sector area for angle α is (α * r^2) / 2. Triangle area with side 2r sin(α/2) and included angle α is (r^2 * sin α) / 2. Subtracting yields the standard formula.

Step-by-Step Solution:

Verification / Alternative check:

Check limits: if α = 0, A = 0; if α = π (half full), A = (r^2 / 2) * (π − 0) = (π r^2)/2; if α = 2π (full), expression gives full circle area (r^2/2)*(2π − 0) = π r^2, which is correct.

Why Other Options Are Wrong:

Options with plus signs overestimate area by adding the triangle; divisors not equal to 2 break the correct sector-triangle relationship.

Common Pitfalls:

Using α in degrees; forgetting to subtract the triangle area; mixing diameter D and radius r inconsistently.

Final Answer:

A = (r^2 / 2) * (α − sin α)