Centroid of a semicircle: The centre of gravity (centroid) of a semicircle of radius r lies at what distance from its base (the diameter), measured along the vertical radius toward the centre?

Introduction / Context:Locating centroids is essential in structural analysis, fluid mechanics (hydrostatic forces), and product design. The semicircle is a standard shape whose centroid location must be memorized or re-derived efficiently for composite-area calculations.

Given Data / Assumptions:

• Planar semicircle of radius r.
• Base is the diameter; distance is measured perpendicularly from this base along the symmetry axis (vertical radius).
• Uniform density and thickness for geometric centroid.

Concept / Approach:By symmetry, the centroid lies on the central vertical radius. Its distance from the diameter is obtained by first principles (area moment integral) or by recalling the standard result. The well-known formula gives the ȳ coordinate from the base as 4r / (3π).

Step-by-Step Solution:Let the diameter lie along the x-axis with origin at the circle's center; the semicircle occupies y ≥ 0.The centroid of the semicircular area measured from the center along y is y_c_from_center = 4r / (3π).Therefore, measured from the base (the diameter) upward along the vertical radius, the same distance is y_c = 4r / (3π).Hence the required distance from the base is 4r / 3π.

Verification / Alternative check:Numerically, 4 / (3π) ≈ 0.424. This places the centroid between the base and the circle's center (at 0.424 r above the diameter), which is physically reasonable because more area mass lies near the base than near the arc.

Why Other Options Are Wrong:

• 3r / 8 = 0.375 r: Too small; not the standard centroid value.
• 8r / 3: Dimensionally wrong (greater than r).
• 3r / 4π ≈ 0.239 r: Too small; not the canonical result.

Common Pitfalls:

• Confusing the centroid from the center versus from the base—the same numeric distance applies here because the base is exactly one radius below the center.
• Mixing arc centroid (of a semicircular arc) with area centroid (of a filled semicircle).

Final Answer:4r / 3π