Last Updated on November 3, 2026 by Maged kamel
Area and CG for a circular sector.
Reference handbook: 10.00 Value for the Area and CG for a circular sector.
A List of common round shapes’ Area and CG values. Our fourth case is a circular sector.
A circular sector is a portion of a Circle closed by two radii and an arc. In our case, we treat it as two radii of length a, with an enclosed angle of 2θ. The external Y-axis passes through the left Point of the circular sector, which has a radius of a. The x-axis bisects the circular sector angle.
Area and CG for a circular sector- select an Area dA.
We have a circular sector with radius a, and we need to find its Area. We can find an internal axis x that divides the circular sector into two similar parts. Because of this symmetry, we expect the center of gravity, Cg, to lie on the X-axis at a distance x from the y-axis. We use the radius of the circular sector as a.
The angle θ is the angle enclosed between the Cg of the strip dA and the X-axis. The angle dθ is the angle enclosed by the strip dA.
The circular sector Area can be found by integrating dA = ∬(ρ*dρ*dθ) from ρ=0 to ρ=a and over an angle equal to dθ. The integration is from θ equals (- θ) to θ=( θ), the final Expression for the Area is dA=1/2*ρ^2*(θ-(-θ)=1/2*a^2(2*θ)=a^2*θ. The steps followed to find the Area for the circular sector are shown in the next slide image.
Area and Cg for a circular sector, first Moment of Area about the Y-axis.
For the Area and Cg for a circular sector 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 the intersection of the two axes.
The first Moment of Area for the small Area dA about the Y-axis is the Product of that Area and the horizontal distance to the Y-axis.
The horizontal distance is x, which equals ρ cos θ. The Moment dMy=dA*(x)=(ρ*dρ*d θ)*(ρ*cos θ). It will be simplified to (ρ^2*dρ*cos θ*dθ).
For the first Moment of Area of the entire circular sector, we will use double integration, since we have to integrate with respect to ρ from ρ=0 to ρ=a. The Second integration is from θ equal to (- θ) to θ equal to (+ θ).
The total Value of the first Moment of Area for the circular sector will be equal to =2*(a^3/3)*(sin(θ-(sin(-θ)=(2*(a^3/3)*sin(θ).
We can find the horizontal distance of the Cg from the Y-axis by dividing the first Moment by the Area Value; this will lead to 2*(a^3/3)*sin(θ))/a^2*θ=(2/3)*(a*sin θ)/θ. The next slide shows the horizontal distance from the Cg to the y-axis.
Area and Cg for a circular sector- first Moment of Area about the X-axis.
For the Area and CG of a circular sector about the X-axis. We have two intersecting axes, X and Y, and we will select a small Area dA with radius ρ centered at the intersection of the two axes.
The first Moment of Area for the small Area dA about the Y-axis is the Product of that Area and the vertical distance to the X-axis.
The horizontal distance is y, which equals 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).
For the first Moment of Area of the entire circular sector, we will use double integration, since we have to integrate with respect to ρ from ρ=0 to ρ=a. The Second integration is from θ equal to (- θ) to θ equal to (+ θ).
The total Value of the first Moment of Area for the circular sector about the X-axis will be equal to =2*(a^3/3)*(-cos(θ-(cos(-θ)=(2*(a^3/3)*0=0.
The vertical distance of the CG to the x-axis will be=0, which means that the CG Point is located along the X-axis. The calculations for the first Moment of Area of the circular sector about the X-axis are shown on the next slide.
We have completed the Area and CG for a circular sector.
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The next Post will cover how to estimate the Area and CG of a circular segment.
This is a very useful site: Engineering statics, open and interactive.