## Area and Cg of a triangle.

### Reference handbook 10.00 values for areas and Cg.

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.

To get the area and Cg for a triangle, we will proceed to divide the triangle into two right-angle triangles, it is required to get the same data as obtained from the FE reference handbook as attached in the next slide image.

### Area and Cg of a triangle-x bar estimation.

We will divide the triangle into two triangles as A1 and A2, the first triangle area can be found as =0.50*base* height, the base length=a, and the common height of the triangle=h.

The Area A_{1}=0.50*a*h. As for the second triangle, The get the x bar for the triangle of area A_{1}, it can be found to be=(2/3)*a from the y axis passing by point a on the left side of the triangle.

The second area is A_{2}, its base=(b-a) and the height=h, so the area A_{2}=0.50(b-a)*h.

While the cg distance for area A2 to the y axis is termed as x bar2 will be =a+(1/3)*(b-a)

The final value for the X- bar for the triangle can be obtained by summing the product of A*x and equating the value to the product of the total area *x bar of the triangle.

The sum of the product of areaA1*X1 bar plus the sum of area 2by x bar2 is shown in the previous slide image.

As we can find out that this product=h/6*(ab+b^2) after we have simplified the terms of multiplications.

The final x bar or the distance between the Cg of the triangle to the y-axis can be found as=(1/3)*(a+b) and matches with the Reference Handbook-10 value of the area and Cg of a triangle.

### Area and Cg of a triangle-y bar estimation.

To get the value of the triangle y bar, we will divide the triangle into two triangles as A1 and A2, and make the necessary calculation to get the y bar for each triangle. The first triangle area can be found as =0.50*base* height, where the base length of the whole triangle =a, while a is the distance from point a to point b’. The common height of the triangle=h.

The Area A_{1}=0.50*a*h.Y1 bar which is the Cg vertical distance to axis x cab be found to be=h/3.

As for the second triangle, The get the y- bar for the triangle of area A_{2}, it can be found to be=(1/3)*h from the x-axis passing by point a on the left side of the triangle.

The second area is A_{2}, its base=(b-a) and height=h. b is the base of the whole triangle ABC. We can divide the product of A2*y2 bar by the area A2.

The area A_{2}=0.50(b-a)*h. while the cg distance to x- axis y- bar-2=h/3.

The final value for The Y- bar for the triangle can be obtained by summing the product of A*y for Areas A1 and A2 and equating the value to the product of the total area *y bar of the triangle.

As we can find out that this product of (A1*y1 bar+A2*y2 bar)=b*h^2/6.The final y bar or the distance between the Cg of the triangle to the x-axis can be found as=h/3. This value matches with the Reference Handbook-10 value of the area and Cg of a triangle.

This is a link for the pdf file used in the illustration of this post.

For a good external reference, please refer to the Centroid of an Area by Integration.

This is the link of the next post will be Area and Cg for a rectangle.