Equilateral-Triangle Cross Sections

From a Base Slice to a Triangle

A solid can be described by the shapes of its cross sections. For this topic, each cross section is an equilateral triangle, and a segment cut from the base region determines one side of that triangle.

The key step is to convert the slice length into a triangle’s area. Once that area is expressed as a function of position, the slicing principle gives the volume by integration. The cross sections may be perpendicular to the \(x\)-axis or to the \(y\)-axis; the integration variable follows the direction in which the slices move through the base.

Area from the Side Length

For an equilateral triangle with side length \(s\), drawing an altitude bisects the opposite side. The resulting right triangle has hypotenuse \(s\), one leg \(s/2\), and altitude \(h\). By the Pythagorean theorem,

Solving for the positive height gives

The area of a triangle is one-half its base times its height. Using side \(s\) as the base and the altitude just found,

The square on the slice length matters: if the slice length doubles, the cross-sectional area becomes four times as large. The factor \(\sqrt{3}/4\) comes from the triangle’s geometry, not from the base region.

Setting Up the Volume

Suppose the base region is bounded by \(y=g(x)\) below and \(y=f(x)\) above for \(a\leq x\leq b\), with \(f(x)\geq g(x)\) throughout the interval. A vertical slice has length \(L(x)=f(x)-g(x)\). If that slice is the side of an equilateral-triangle cross section, its area is \(\frac{\sqrt{3}}{4}[L(x)]^2\). The volume is the integral of the cross-sectional area:

The same reasoning applies to horizontal slices. If a horizontal slice at height \(y\) runs from \(x=p(y)\) to \(x=q(y)\), with \(q(y)\geq p(y)\) on \(c\leq y\leq d\), its length is \(L(y)=q(y)-p(y)\). The volume is then

The bounds and slice length must describe the same direction: vertical slices use \(dx\), while horizontal slices use \(dy\). Lengths are nonnegative on the stated intervals, and the integral adds the areas of the cross sections across the solid.

Worked Examples

A Reliable Setup and a Common Pitfall

Before integrating, identify the direction of the slices and the segment in the base that determines each triangle’s side. The cross-sectional area uses that segment’s length, not a distance measured in some other direction. Then use the area formula and integrate over the interval or range of positions.

A common mistake is to use the slice length as the triangle’s height. For a side length \(s=2\), this incorrect calculation treats the triangle as having base \(2\) and height \(2\), giving area \(\frac12(2)(2)=2\). But an equilateral triangle of side \(2\) has height \(\frac{\sqrt{3}}{2}(2)=\sqrt{3}\), so its area is

The correction is to remember that the base slice specifies the side length. The triangle’s perpendicular height must be calculated from that side; it is not generally equal to it.

Why the Side-to-Area Formula Works

The area formula is not an extra assumption about the solid. It follows from the geometry of each planar cross section. The altitude divides an equilateral triangle into two right triangles. With hypotenuse \(s\) and shorter leg \(s/2\), the Pythagorean theorem forces the altitude to be \(\sqrt{3}s/2\). Substitution into the triangle area formula then forces the cross-sectional area to be \(\sqrt{3}s^2/4\).

As the slice position changes, the base segment length may change, so the area changes too. Integrating those areas across the full interval adds the cross sections to obtain the volume. This is why it is essential to square the correct slice length before integrating.