Question:
Locate the centroid of a semicircle arc about its horizontal base, which is symmetric about its vertical axis and diameter of the arc is \(18\pi\) cm.
Locate the centroid of a semicircle arc about its horizontal base, which is symmetric about its vertical axis and diameter of the arc is \(18\pi\) cm.
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Remember the centroid formulas:
For a semicircular arc,
\[
\boxed{\bar{y}=\frac{2R}{\pi}}
\]
For a semicircular area,
\[
\boxed{\bar{y}=\frac{4R}{3\pi}}
\]
Do not confuse the centroid of an arc with that of an area.
Updated On: Jul 23, 2026
- \(36\) cm
- \(24\) cm
- \(18\) cm
- \(12\) cm
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Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
The centroid of a semicircular arc lies on its axis of symmetry at a distance
\[
\boxed{\bar{y}=\frac{2R}{\pi}}
\]
from its diameter (horizontal base), where \(R\) is the radius of the semicircular arc.
Step 1: Determine the radius of the semicircular arc. Given that the diameter is \[ 18\pi\ \text{cm}, \] therefore, \[ R=\frac{18\pi}{2}=9\pi\ \text{cm}. \]
Step 2: Use the centroid formula. The distance of the centroid from the horizontal base is \[ \bar{y}=\frac{2R}{\pi}. \] Substituting \(R=9\pi\), \[ \bar{y} = \frac{2(9\pi)}{\pi} = 18\ \text{cm}. \] Hence, \[ \boxed{\bar{y}=18\ \text{cm}.} \] Therefore, the correct option is \[ \boxed{(C)\;18\ \text{cm}.} \]
Step 1: Determine the radius of the semicircular arc. Given that the diameter is \[ 18\pi\ \text{cm}, \] therefore, \[ R=\frac{18\pi}{2}=9\pi\ \text{cm}. \]
Step 2: Use the centroid formula. The distance of the centroid from the horizontal base is \[ \bar{y}=\frac{2R}{\pi}. \] Substituting \(R=9\pi\), \[ \bar{y} = \frac{2(9\pi)}{\pi} = 18\ \text{cm}. \] Hence, \[ \boxed{\bar{y}=18\ \text{cm}.} \] Therefore, the correct option is \[ \boxed{(C)\;18\ \text{cm}.} \]
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