3-How to find the area and Cg for a semi-circle?

Last Updated on March 7, 2026 by Maged kamel

Area and Cg for a semi-circle.

Reference handbook 10.00 value for the area and Cg for a semi-circle.

There is a list of common round-shaped areas and CG values. Our third case is a semicircle. The external axis Y is tangent to the semicircle, which has a diameter of 2a. The x-axis passes through the center of the semicircle.

Area and Cg for a semi-circle- select an area dA.

We have a semi-circle with radius a, which requires getting the area and the CG. We can find that there’s an internal axis Y that divides the semicircle into two similar parts. Due to that symmetry, we expect that the Cg or the center of gravity will be located along the y-axis at a certain distance y from the external axis x. We have taken the radius of the semicircle to be r.

We need a small area dA to estimate the first moment of the area and obtain the total area of the semi-circle.

Area and Cg for a semi-circle, first moment of area about the Y-axis

For the area and Cg for a semi-circle about the y-axis. We have two intersecting axes, X and Y, and we will select a small area dA with radius ρ centered at their intersection.

We will differentiate between two cases: the first case, which we are facing, where the Y axis is the axis of symmetry; for this case, we will take an angle θ between the Y axis and the line to the Cg of the strip dA. For integration, the angle θ will be from -π/2 to π/2.

The second case, where the X axis is the axis of symmetry, for this case, we will take an angle θ between the X axis and the line to the Cg of the strip dA. For integration, the angle θ will be from -π/2 to π/2, and x bar will not be zero.

The first moment of area for the small area dA about the Y-axis is the product of that area by the vertical distance to the Y-axis. The area of the strip equals(ρ*dρ*d θ).

The horizontal distance is x, which is equal to ρ*sin θ. The moment dMx=dA*(y)=(ρ*dρ*d θ)*(ρ*sin θ). It will be simplified to (ρ^2*dρ*sin θ*dθ).

We will use double integration to compute the first moment of area for the entire semi-circle, since we have to integrate ρ from ρ=0 to ρ=r. The second integration is for the angle dθ, from θ = -π/2 to θ = π/2, or from -90 degrees to +90 degrees. dA*y=∬(ρ^2*dρ*sin θ*dθ).

The value of integration about the Y-axis is zero, which indicates that the Cg is the center of the semicircular point.

Use a vertical strip to obtain the first moment of area about the Y-axis

We can use a vertical strip as a second option to the distance X bar for the semi-circle. The height of the strip is y, and the thickness is dx, and the Cg vertical distance is y/2.

The area of the strip equals (dx*y) while the first moment of area about the Y axis is equal to x*dx*y. We can use the expression of Y=sqrt(r^2-x^2) where r is the semicircle radius, and u is equal to y^2, the value of du equals =-2x, then X=-du/2.

Back to the first moment equation, it will be equal to -u*du/2. The strip will move from -r to r, which is from u=0 to u=0. The first moment of area about the Y-axis will be zero, as shown in the next slide.

Divide A*y/A to get Y bar will lead to y-bar Cy=2*(r^3/3)/0.5*π*a^2=4a/3*π. This includes the fact that the Cg lies on the X-axis.

Area and Cg for a semi-circle, first moment of area about the X-axis.

For the area and Cg for a semicircle about the X-axis, the first moment of area dA about the X-axis is dMy=dA*(y)=(ρ*dρ*d θ)*(ρ*cos θ). It will be simplified to (ρ^2*dρ*cos θ*dθ).

We will use double integration to compute the first moment of area for the entire semi-circle, since we have to integrate ρ from ρ=0 to ρ=r. The second integration is for the angle dθ, from θ = -π/2 to θ = π/2. dA*y=∬(ρ^2*dρ*cos θ*dθ).

The value of the first integration is equal to ∫ρ^2*dρ from zero to r=1/3*ρ^3, after substitution, will lead to 1/3*(a^3-0)=a^3/3, while  ∫(cos θ*dθ)=sin(θ), after substitution from -π/2 to π/2. The value will be (+2r^3/3)

We have estimated the area of the semi-circle to be 0.5*π*r^2, as shown in the next slide.

The CG distance is Y bar = 4r/3, as shown on the next slide.

Use a vertical strip to obtain the first moment of area about the X-axis.

We can use a vertical strip as an alternative to the distance y-bar for the semicircle. The height of the strip is y, and the thickness is dx, and the Cg vertical distance is y/2.

The area of the strip equals (dx*y) while the first moment of area about the X axis is equal to y*dx*y/2=y^2*1/2*dx. Use the expression for Y^2=r^2-x^2. We perform the integration from -r to +r.

Back to the first moment equation, it will be equal to -u*du/2. The strip will move from -r to r, which is from u=0 to u=0. The first moment of area about the Y-axis will be zero, as shown in the next slide.

We have completed the subject of the area and CG for a semi-circle.

The PDF of this post’s content can be viewed or downloaded from the following document.

The next post will be on estimating the area and Cg for a circular sector.

This links to a very useful site: Engineering statics open and interactive.