Composite canal section: For a triangular lined canal with a circular bottom of radius R, where the straight sides make an angle θ with the horizontal, what is the wetted perimeter P?

Introduction / Context:

Compound canal sections often combine a curved (circular) bottom with straight side walls. The wetted perimeter P is crucial in hydraulic computations (e.g., Manning–Strickler) because it affects the hydraulic radius and therefore the conveyance capacity of the lined channel.

Given Data / Assumptions:

• Circular invert (bottom) of radius R.
• Straight side plates/lips making angle θ with the horizontal.
• Section is symmetric about the vertical centreline.
• Wetted perimeter comprises the circular arc plus two identical straight side lengths.

Concept / Approach:

Let the circular invert subtend an angle 2θ at the centre. The circular arc length is 2 * R * θ. Each straight side rises from the arc tangent point at angle θ to the horizontal; the wetted length of one side equals R * tan θ (from right-triangle geometry at the tangent point). Total side contribution is 2 * (R * tan θ). Hence, P = 2Rθ + 2R tan θ = 2R(θ + tan θ).

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: θ is dimensionless; tan θ is dimensionless; R carries length. Therefore P has length units as required.

Why Other Options Are Wrong:

• R(θ + tan θ): Misses the symmetry factor 2.
• Expressions with cos θ: Do not represent the side lengths deriving from the tangent geometry; tan θ is the correct trigonometric relation.
• None of these: Incorrect because a standard closed-form exists, P = 2R(θ + tan θ).

Common Pitfalls:

• Mistaking θ as measured to the vertical; the problem states angle with the horizontal.
• Omitting the factor 2 for the two symmetric sides and the full arc.

Final Answer:

2R(θ + tan θ)