Last Updated on September 10, 2026 by Maged kamel
7- Easy a
Area and Cg for a Rectangle.
Reference handbook 10.00 value for Area and Cg in the x-direction.
To find the Area and cg for a Rectangle, we integrate an infinitesimal Area of width dx and height dy about the external axes. We need to get the same data as in the FE reference handbook, as shown in the image on the next slide, where Xc = b/2. Xc; we call it X-bar.
This picture from the FE reference handbook shows the Area and Cg for different plane shapes.
Area and Cg for a Rectangle in the x-direction (x-bar).
Our new topic is estimating the Area and Cg of a Rectangle. We will do this by integrating a strip of Area (dx*dy), and summing these tiny areas gives the total Area.
The summation of this Area’s first Moment about the Y-axis will give the first Moment of the Area of the whole Rectangle about the Y-axis.
But the double integration includes two operations: first, integrate dy from y=0 to y=h to create a rectangular strip; then, take the Moment of that strip about y to get the first Moment of Area of the Rectangle about the Y-axis.
The detailed calculation for the first Moment of Area for the rectangular shape is presented in the next slide image. The final value of the first Moment of Area about the y-axis will be h*b^2/2.
The final value of x-bar will be obtained by dividing the value of the first Moment of Area by the value of the rectangular Area.
The Area of the Rectangle is the Product of the base and the height: A = bh. The final value of the X-bar=b^2h/2/(0.50bh)=b/2, or half the breadth; the result will match the Reference handbook value given.
Reference handbook 10.00 value for Area and Cg in the y- direction.
To get the Area and Cg for a Rectangle, we will integrate an infinitesimal Area of (dx) width and dy height, about the external axes. We need to get the same data as in the FE reference handbook, as shown in the image on the next slide, where yc = h/2. We call yc y-bar.
Area and Cg for a Rectangle in the y- direction-(y-bar).
Now we will estimate the first Moment of Area about the x-axis. We do this by integrating a strip of Area (dx*dy), and summing these tiny areas gives the total Area.
The process uses double integration; the full details are shown in the next slide images.
Likewise, the summation of this Area’s first Moment about the x-axis will give the first Moment of the Area of the whole Rectangle about the x-axis, which will be =b*h^2/2.
The final value of the y-bar will be obtained by dividing the value of the first Moment of the Area by the value of the rectangular Area.
The Area of the Rectangle is the Product of the base and the height: A = b*h. The final value of y- bar=h^2*b/2/(0.50*b*h)=h/2, or half the total height of the Rectangle; the result will match the Reference handbook value given. Thus, we have completed the final values of Area and Cg for a Rectangle.
Another alternative is to consider a vertical strip of breadth dx and height h and integrate about the y-axis; this is a single integration, and we get the same result for x-bar.
Another alternative is to consider a horizontal strip of breadth b and thickness dx and integrate about the x-axis; this is a single integration, and we get the same result for y-bar.
You can view or download the PDF for this post via Area and CG for a Rectangle.
For a good external reference, see Centroid of an Area by Integration.
The next post will be Area and Cg for a Trapezium.