## Area and Cg for a circular sector.

### Reference handbook 10.00 value for the area and Cg for a circular sector.

There is a list of the common round shapes area and CG value. Our fourth case is the case of a circular sector.

You can click on any picture to enlarge, then press the small arrow at the right to review all the other images as a slide show.

The circular sector is a portion of a circle that is closed by two radii and an arc. Our case will be treated as two radii of length a and the enclosed angle is 2**θ**.

The external axis Y is passing by the left point of the circular sector which has a radius value of a. The x-axis is Bisecting the angle of the circular sector.

### Area and Cg for a circular sector- select an area dA.

We have a circular sector with radius a, for which it is required to get the area and CG. We can find that there’s an internal axis x that divides the circular sector into two similar parts. Due to that symmetry, we expect that the Cg or the center of gravity will be located along the X-axis at a certain distance x from the external axis Y. We have used the radius of the circular sector as equal to a.

The angle **θ** is the angle enclosed between the Cg of the strip dA with the X-axis. The angle d**θ** is the enclosed angle by the strip dA.

The circular sector area can be found from the integration of dA= ∬(**ρ***d**ρ***d**θ**) from **ρ**=0 to **ρ**=a and enclosed by an angle equal to d**θ**. 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 the area for the circular sector is 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 Y-axis. We have two intersecting axes X&Y, we will select a small area dA that has a radius of ρ from the intersection of the two axes X and Y.

The first moment of area for the small area dA about the Y-axis is the product of that area by the horizontal distance to the Y-axis.

The horizontal distance is x, which is equal to **ρ***cos **θ**. The moment dMy=dA*(x)=(**ρ***d**ρ***d **θ**)*(**ρ***cos **θ**). It will be simplified to (**ρ**^2*d**ρ***cos **θ***d**θ**).

For the first moment of area for the whole circular sector, we will use double integration since we have to integrate for **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 details of the horizontal value of the horizontal distance of the Cg to the y-axis is shown in the next slide image.

### Area and Cg for a circular sector-first moment of area about the X-axis.

For the area and Cg for a circular sector about X-axis. We have two intersecting axes X&Y, we will select a small area dA that has a radius of ρ from the intersection of the two axes X and Y.

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 X-axis.

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

For the first moment of area for the whole circular sector, we will use double integration since we have to integrate for **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 details of the calculations for the first moment of area for the circular sector about the X-axis are shown in the next slide image.

We have completed the subject of the area and Cg for a circular sector.

The next post will be on how to estimate the area and Cg for a circular segment.

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