Breif description of Post 6-inertia.

6- Easy steps to find the area and Cg of a triangle.

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.

Reference handbook values of the area

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.

Area and Cg of a triangle

The Area A1=0.50*a*h. As for the second triangle, The get the x bar for the triangle of area A1, 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 A2, its base=(b-a) and the height=h, so the area A2=0.50(b-a)*h.

Area and cg of triangle -2

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

The equation to get x bar for a triangle.

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 value X bar value for a triangle.

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 A1=0.50*a*h.Y1 bar which is the Cg vertical distance to axis x cab be found to be=h/3.

Area and y bar for the first triangle.

As for the second triangle, The get the y- bar for the triangle of area A2, 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 A2, 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.

Area and y bar for the second triangle.

The area A2=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.

Area and final y bar value for a 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.

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