9- Easy approach to Area and CG for a Parallelogram.

Last Updated on September 21, 2026 by Maged kamel

9- Easy a

Area and Cg for a Parallelogram.

Reference handbook 10.00: Value for Area and Cg in the x-direction.

To find the Area and Cg for a Parallelogram, divide it into a Rectangle and two triangles, and take the first Moment of Area about a vertical axis y passing through the external edge point about the external axes. This will give the x-bar of the Parallelogram.

For y-bar, follow the same process, but take the first Moment of Area about an external axis passing through the base, which is then called the x-axis. We need to obtain the same data as in the FE reference handbook, as shown in the image on the next slide, for case #No.6.

Area and CG for a Parallelogram in the x-direction.

To get Area and CG for a Parallelogram in the x-direction. We will check the base b and height h, and then we divide it into the following shapes:

1-A left Triangle of base (b₁cos θ) and a height of h;; itsAreaa A2=1/2*( b₁cos θ)*h;; its CG is apart from the y-axis by a distance x1=(2/3*(b₁cos θ).

2-A Rectangle of base (b-b₁*cos θ) and height h; its Area is A1=(b-b1*cos θ)*h, and its CG is a distance x2=1/2*(b+b₁*cos θ) from the y-axis.

3-A right triangle of base b₁*cos θ and a height of h, its area A3=1/2*b₁*cos θ*h, its CG is apart from the y-axis by a distance x3=(b+1/3*b₁*cos θ).
We will simplify the Expression by adjusting the terms; the Area will be b*h, as a skewed Rectangle.

The Area of the Parallelogram is the Area of the left triangle+area of the Rectangle + Area of the right Triangle. The next slide images show Rectangles for these shapes.

x₁,x₂, and x₃ values are shown in the next slide image.x₁ is the first Triangle Cg distance to the y-axis. Similarly, x₂ is the Cg distance for the Rectangle shape from the y-axis. x₃ is the x distance for the third shape from the Y-axis.

The first Moment of Area for these three shapes will be = the first Moment of Area for the Parallelogram about the Y-axis.

The final Value for Area and Cg of a Parallelogram in the x-direction.

We will have the final Expression for the distance of the Parallelogram’s Cg about the y-axis, which is X-bar. The next example will show the same Expression, but with a- instead of a.

Area and Cg for a Parallelogram in the y-direction.

The Parallelogram with base b and height h is divided into the following shapes:

1-A lefTrianglele of base (b₁cos θ) and a height of h,;its Area A2=1/2*( b₁cos θ)*h,;its CG is apart from the x-axis by a distance y1=(1/3*h).
2-A Rectangle of base (b-b₁*cos θ) and a height h; its area A1=(b-b1*cos θ)*h; its CG is apart from the x-axis by a distance y2=1/2*h.
3-A right Triangle of base b₁*cos θ and a height of h; its Area A3=1/2*b₁*cos θ*h; its CG is apart from the X-axis by a distance Y3=2/3h.

The Area of the Parallelogram is the Area of the left triangle+area of the Rectangle + Area of the right Triangle. The details of these shapes are shown in the images on the next slide.

After adjusting the terms, we get Y bar = h/2. The Expression in terms of a will use h = b1*cos θ.

To get the Expression for the Rectangle, consider θ = 90 degrees. Rectangle Moment of Inertia Ix for a Parallelogram; please find the Link.

You can view or download the PDF for this Post via the Area and CG for a Parallelogram document.

This links to the previous Post. We have estimated the Area and CG for a Trapezium.

Engineering Statics Open and Interactive provides a helpful Link for areas to view the CG.