## 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 the common round shapes area and CG value. Our third case is the case of a semi-circle.

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 external axis Y is tangents to the semi-circle which has a diameter value of 2a. The x-axis is passing by the center of the semi-circle.

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

We have a semi-circle with radius a, for which it is required to get the area and CG. We can find that there’s an internal axis Y that divides the semi-circle 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 used the radius of the semi-circle as equal to r.

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

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

For the area and Cg for a semi-circle 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 X-axis is the product of that area by the vertical distance to the X-axis. 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 details of the horizontal and vertical distances of the strip dA are shown in the next slide image.

The vertical 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 semi-circle, we will use double integration since we have to integrate for **ρ** from **ρ**=0 to **ρ**=r. The second integration os for the angle d**θ** from **θ** equals to zero to d**θ** equal to π or 180 degrees. dA*y=∬(**ρ**^2*d**ρ***sin **θ***d**θ**).

This is the value of the first integration, ∫**ρ**^2*d**ρ** from zero to r will be equal to 1/3***ρ**^3 after substitution will lead to 1/3*(r^3-0)=r^3/3. While for the second integration ∫(sin **θ***d**θ**)=-cos(**θ**), after substituting from zero to π. The value will be (- (cos(π)- cos(0))=(-(-1)-(1)=+2. The total value of the first moment of area for the semi-circle will be equal to =2*(r^3/3).

But the semi-circle area can be found from the integration of dA= ∬(**ρ***d**ρ***d**θ**) from **ρ**=0 to **ρ**=a and for d**θ** from **θ** equals zero to d**θ**=π, the final expression for the area is dA=**ρ**^2*0.5*(**θ**), substitute to get A=0.50*(r^2-0)*(π-0)=0.50*π*r^2.

The details of the calculations for the first moment of area for the semi-circle about the X-axis and the value of the area are shown in the next slide image.

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

### 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. The first moment of area dA about the y’-axis is 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 semi-circle, we will use double integration since we have to integrate for **ρ** from **ρ**=0 to **ρ**=r. The second integration os for the angle d**θ** from **θ** equals to zero to d**θ** equal zero to π or 180 degrees. dA*y=∬(**ρ**^2*d**ρ***cos **θ***d**θ**).

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

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

The Cg distances are x bar=0 and y bar=4r/3* and are shown in the next slide image.

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

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

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