4- Easy steps for X-bar for a right-angle case 2.

Last Updated on September 10, 2026 by Maged kamel

How to determine the x-bar for a right-angle case 2?

For more information about the difference between case-1 and case-2, please refer to post-2.

Using a horizontal strip to find x-bar for right-angle case 2.

We will start by using a horizontal strip to find the value of the X-bar for a right-angle case 2,2, or the CG horizontal distance to the y-axis.
We have X and Y axes respectively and the base of the triangle.

We have line AB with length b; the rise of the triangle is h; and the inclined portion is AC. The equation is y = m*x, where m (the slope) equals +h/b, and the intersection with the y-axis is 0. The previous slide shows the following four steps.

Integrate the horizontal strip to find the Area of the right-angle triangle.

The Area of the triangle is the sum of all the tiny horizontal strips, which we express as an integral from y=0 to y=h, considering the strip moving vertically.
Since the strip width is (b-x) and its height is dy, we are going to use the relation between y and x as derived from the equation of line BC.

We will estimate the Area dA as x*dy. Since integration is in the vertical direction, we will omit the X Expression by substituting its value in terms of y, the x value (b*y/h). Proceed with the integration; we get the final Area = 0.50 b*h, which is the known formula for the Area of a right-angle triangle: 0.5*base*height.

Integrate the horizontal strip to get the first Moment of Area about the Y-axis.

The Expression of dA*x-strip will be represented by the first Moment of Area about the y-axis, where the x-strip is the horizontal distance from the Cg of the strip to the y-axis.

The Expression of dA*x-strip is shown in the next slide image, and integration will be carried out in the vertical direction from y=0 to y=h.

We notice that x strip from the cg of the strip to the y-axis=x+(b-x)*0.50=0.50(b+x).

The final A*x bar represents the Product of the total Area * the horizontal CG distance from the y-axis, which will be found as, in our case=b^2*h/3, where b is the triangle base and h is the height.

X bar for a right-angle case:: 22,, final step.

Continue the estimation of the integration of the Product of A *x bar; the full details are shown in the next slide image.

The x-bar value will be obtained by simply dividing the first Moment of Area /Area; we will get x-bar for a right angle=2*b/3 or two-thirds of the base width.

Using a vertical strip to get X-bar for a right-angle case-2.

Another approach is to use a vertical strip to get the value of the X-bar, or the CG horizontal distance to the y-axis.
We have line AB with length b; the rise of the triangle is h; and the inclined portion AC has the equation y = mx + C, where m is the slope (h/b), and the y-intercept is 0.

The width of the strip =dx, and its height equals y. The Area of that strip is dA = y*dx. Since we are integrating in the x-direction, we will omit the Expression of y by substituting its value in terms of x. The next image shows the procedure. The Area is 0.50*b*h, the same result obtained earlier using the horizontal strip.

Integrate the vertical strip to find the first Moment of Area about the Y-axis.

The Expression of dA*x-strip will be represented by the first Moment of Area about the y-axis, where x-strip is the horizontal distance from the Cg of the strip to the y-axis. The Expression of dA*x-strip is shown in the next slide image, and integration will be carried out in the vertical direction from x=0 to x=b.

The final A*x bar represents the Product of the total Area * the horizontal CG distance from the y-axis, which will be found as, in our case=b^2*h/3, where b is the triangle base and h is the height.

X bar for a right-angle case- 2nd final step.

X bar value will be obtained by simply dividing the first Moment of Area /Area; we will get x bar for a right angle=b/3 or two-thirds of the base width.

You can view or download the PDF for this post via X-bar for a right-angle triangle case 2.

You can view or download the PDF for this post via X-bar for a right-angle triangle case 2.

This is a link to a good external reference: The Engineering Toolbox.

Here is the link to the next post: y-bar for a right-angle case 2.