What is the centroid of a semicircle?

Short Answer:

The centroid of a semicircle is the geometric center or average position of all the points within the semicircular area. It lies along the axis of symmetry (the central vertical line of the semicircle) and at a certain distance from the flat base. The centroid does not coincide with the center of the circle but is located closer to the base. For a semicircle of radius r, the distance of the centroid from the base is given by the formula:

This value approximately equals 0.424r.

Detailed Explanation :

Centroid of a Semicircle

In Engineering Mechanics, the centroid represents the geometric center of an area or body, showing the mean position of all its points. For a semicircle, the centroid lies on its axis of symmetry — a line that divides the shape into two equal halves — and at a distance from the base that depends on the radius.

Unlike a full circle, where the centroid coincides with the center, a semicircle’s centroid is not at the center of the circle because part of the area (the upper half) is missing. The centroid moves downward toward the base due to the uneven distribution of the area. This makes finding the centroid of a semicircle an essential problem in geometry and engineering applications such as beam design, pressure distribution, and center of gravity calculations for curved surfaces.

Definition

The centroid of a semicircle can be defined as:
“The point on the axis of symmetry of a semicircle at which the entire area of the semicircle may be assumed to be concentrated.”

It is the point where the shape balances perfectly if it were made of a uniform material. The centroid always lies on the axis of symmetry of the semicircle and above the base, but not at its geometric center.

Position of the Centroid

Let the semicircle have:

  • Radius = r
  • Flat base = diameter = 2r

The centroid lies along the vertical axis of symmetry (the line passing through the center of the circle perpendicular to the base).

The distance of the centroid from the flat base is given by:

This formula gives the vertical location of the centroid measured from the flat base toward the center of curvature.

In decimal form,

Hence, the centroid lies approximately 0.424 times the radius above the base line.

Example:
If the radius of the semicircle is 100 mm, then

Therefore, the centroid is 42.4 mm above the flat base.

Derivation of Centroid of a Semicircle

To find the exact position of the centroid, we use the integration method based on calculus principles.

Let’s consider a semicircle of radius r, lying above the x-axis with its center at the origin (0,0).

The equation of the circle is:

For the upper semicircle,

Now, we can find the centroid using the general formula for the centroid of an area:

Since the semicircle is symmetrical about the y-axis, its x-coordinate of the centroid is zero, i.e.,

To calculate , we consider a thin horizontal strip of width dx and height y at a distance x from the y-axis.

The elemental area of the strip is:

Substituting in the centroid formula:

After integration and simplification (using standard results from calculus), we obtain:

This formula gives the distance of the centroid from the flat base of the semicircle.

Location of Centroid with Respect to the Circle

The centroid of the semicircle lies below the center of the full circle, closer to the flat base.

The distance from the center of the circle to the centroid is:

 

This is approximately equal to 0.576r.

Hence, the centroid lies about 0.576r below the center of the circle.

Geometrical Construction of the Centroid

The centroid can also be found graphically for a semicircular lamina using the following simple method:

  1. Draw the semicircle with its base along the x-axis.
  2. Locate the center O and draw the axis of symmetry (vertical line through O).
  3. Measure the radius r.
  4. Mark the centroid G on the axis of symmetry at a distance of 4r / 3π from the flat base.

This point G represents the centroid of the semicircular area.

Physical Meaning

If a semicircular plate of uniform thickness and material is made, the centroid represents its center of gravity. This means if the plate is supported at that point, it will balance horizontally without tilting.

In physical terms:

  • The centroid is the average location of the area.
  • It is the point at which the whole weight of the semicircular plate can be assumed to act.

Practical Applications

The concept of the centroid of a semicircle is widely used in various branches of mechanical and civil engineering, such as:

  1. Structural Engineering:
    To find the neutral axis and moment of inertia of beams with semicircular cross-sections.
  2. Mechanical Engineering:
    In the design of curved components such as pipes, flywheels, and arches.
  3. Hydraulics:
    To determine the center of pressure and stability of curved surfaces in contact with fluids.
  4. Manufacturing:
    Used in designing gears, pulleys, and other components having semicircular profiles.
  5. Robotics and Aerospace:
    Helps in locating the balance point of curved plates and segments.

Example Problem

Given:
A semicircular plate has a radius of 150 mm.
To find: Distance of centroid from the flat base.

Solution:

 

Therefore, the centroid lies 63.7 mm above the base.

Key Observations

  • The centroid lies along the axis of symmetry.
  • It always lies between the base and the center of the circle.
  • It depends only on the radius of the semicircle, not on its thickness or material.
Conclusion

In conclusion, the centroid of a semicircle is the point on its axis of symmetry located at a distance of (4r / 3π) from the flat base. It represents the geometric center or balance point of the semicircular area. The centroid is essential in determining structural balance, stress distribution, and mechanical stability in curved surfaces and circular sections. Understanding its position helps engineers and designers accurately calculate moments, equilibrium, and pressure centers for various engineering applications.