Triangular Cross-Sections From Base-Region Chords

From a Chord to a Triangular Cross-Section

A solid can be described by the shapes of its slices. In this tutorial, each slice is a triangle, and one dimension of that triangle comes from a chord of a planar base region. The chord length is not necessarily the triangle’s full perimeter or a side visible in a three-dimensional drawing: it is the segment cut out of the base region by the plane of the slice.

The central task is to connect that chord length to the triangle’s area. Once the cross-sectional area is written as a function of the slice position, integrating it gives the volume. The geometric condition on the triangles matters: “right triangle” alone may not determine an area from a chord, whereas a specified altitude-to-base ratio does.

Finding the Chord Length

For vertical slices, a chord runs from the lower boundary of the base region to its upper boundary. If those boundaries are \(y=g(x)\) and \(y=f(x)\), with \(f(x)\geq g(x)\) on the interval, then

For horizontal slices, the chord runs from the left boundary to the right boundary. If those boundaries are \(x=u(y)\) and \(x=v(y)\), with \(v(y)\geq u(y)\), then

These lengths are determined by the planar base region. The triangle’s altitude is a separate piece of information, supplied by the cross-section’s shape condition. Keep the two roles distinct: the chord provides the triangle’s base (or, in some right-triangle specifications, its hypotenuse), while the shape condition determines how the other dimensions relate to it.

Area Formulas for Triangular Slices

The area of any triangle is one-half its base times its perpendicular altitude. Therefore, if the chord is the chosen base and the altitude is \(h(x)\), then \(A(x)=\frac12 b(x)h(x)\). A ratio condition often makes this especially simple.

A common right-triangle condition says that the chord is one leg and the perpendicular leg has a specified length or ratio. In that case the perpendicular leg is the altitude to the chord, so the triangle area formula applies directly. For example, if both legs have length \(b(x)\), then \(h(x)=b(x)\) and \(A(x)=\frac12(b(x))^2\).

A different condition may make the chord the hypotenuse. Knowing only that a triangle is right-angled and that its hypotenuse has length \(b(x)\) does not determine its area: the other two sides can vary. An additional condition is needed, such as a specified altitude to the hypotenuse or a specified relationship between the legs. For an isosceles right triangle with chord as hypotenuse, the two legs are equal, and the area can be determined from the chord; this case is worked below.

Why Integrating the Areas Gives Volume

Suppose vertical slices run across \(a\leq x\leq b\), and \(A(x)\) is the area of the triangular cross-section at position \(x\). Divide the interval into short widths \(\Delta x\). Over a sufficiently short interval, a thin part of the solid is approximately a prism with volume \(A(x_i^*)\Delta x\), where \(x_i^*\) is a sample point in that interval. Adding the thin volumes gives an approximation to the whole volume:

As the widths become smaller, these sums approach the definite integral. Thus, when the cross-sectional area is integrable, the volume is

For horizontal slices over \(c\leq y\leq d\), the corresponding formula is \(V=\int_c^d A(y)\,dy\). The integration variable describes the direction in which the slices are stacked. Since area has units of length squared and \(dx\) or \(dy\) has units of length, the integral has units of volume.

Worked Examples

A Reliable Setup

Before integrating, make each geometric choice explicit. A short setup sequence helps prevent a correct integration from answering the wrong geometric question.

Common Pitfall: Omitting the One-Half

A frequent error is to multiply the chord by the altitude but forget the factor \(\frac12\). For example, if the chord is \(b=2\) and the altitude is \(h=\frac32b=3\), the incorrect calculation \(A=bh=2(3)=6\) is not the area of the triangle. The correct calculation is

The same error changes the whole volume integral: it doubles every cross-sectional area and therefore doubles the volume. A quick check is to compare the proposed triangular area with the area \(bh\) of a rectangle with the same base and altitude. The triangle’s area must be half of that rectangle’s area.

Another important check concerns right triangles. If the chord is the hypotenuse, a right angle alone does not fix the altitude or the area. A definite area formula requires additional information, such as the isosceles condition in the third example or a specified altitude-to-hypotenuse ratio.

Why the Geometry Matters

The chord is determined by the base region, but the cross-sectional shape determines how much area is built above that chord. The same base region can therefore produce different volumes under different triangle conditions. For instance, changing the altitude-to-base ratio changes the coefficient in \(A(x)=\frac{r}{2}(b(x))^2\); changing which segment the chord represents can require a different geometric calculation altogether.

Keeping the base region, slice direction, chord, and triangle condition separate makes the method dependable. It also provides useful checks: a zero-length chord should give a zero-area cross-section, a positive chord and positive altitude should give positive area, and integrating area over length should produce volume.