5-Y-bar for a right-angle case-2

Last Updated on September 10, 2026 by Maged kamel

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

Our subject is to determine the y-bar for a right-angle case-2, but first, what are the differences between case-1 and case-2?

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

Using a horizontal strip to get y-bar for a right-angle case-2.

We will start by using a horizontal strip to get the value of the y-bar, or the Cg vertical distance to the y-axis.
We have the X and Y axes, respectively, and the triangle’s base.

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 the slope is m = +h/b, and the y-intercept is C = 0.

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 can express as an integral from the start, y=0, to the end, y=h, considering the strip moving vertically.

Since the strip width is dy and it is at a distance y from the x-axis. We are going to use the relation between y and x as derived from the equation of line BC. We estimate the Area dA as (b-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 aArea= 0.50*b*h, the known formula for the Area of a right-angle triangle: half the base times the height.

Integrate the horizontal strip to get the first-moment Area about the x-axis.

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

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

The final A*y bar represents the Product of total Area * the vertical CG distance from the y-axis, which will be found as, in our case=b*h^2/6, where b is the triangle base while h is the height. The y-bar value will be obtained by simply dividing the first Moment of Area /Area.

We estimate the Product of dA by y; it will be equal to (b-b/h*y)dy*y. We integrate from y=0 to y=h. Finally, the x-bar value is =b*h^2/6.

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

Another approach is to use a vertical strip to find y-bar, or the Cg vertical distance to the X-axis.
We have X and Y axes respectively and the base of the triangle. We have line AB with the length of b, the rise of the triangle is=h, and the inclined portion AC, equation: y =mx+C m which is a slope is equal to +-h/b *x, and the intersection with y-axis =0. The width of the strip =dx and its height=y.

The strip Area 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 X-axis.

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

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

The next slide shows the integration details. The final A*y bar for a right-angle case-2 represents the Product of the total Area * the vertical CG distance from the x-axis, which will be found as,, in our case=b*h^2/6, where b is the triangle bas and h is the height.

Y-bar for a right-angle case 2-2 will be obtained by simply dividing the first Moment of Area /Area; we will get Y-bar for a right angle=h/3 or one-third of the opposite side height.

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

This is a link to a good external reference; please refer to the Centroid of an Area by Integration.

This is a link to the next post: Area and Cg of a triangle.