From a Base-Region Chord to a Cross-Section
A solid can be described by its base region and by the shape of each cross-section perpendicular to a chosen axis. At a location along that axis, the base region supplies a chord: the segment joining the two boundaries of the base at that location. If that chord is a side of an equilateral-triangle cross-section, its length determines the triangle’s area, and the areas of all the slices determine the volume.
The key is to keep the geometric quantities distinct. The chord is a length in the base plane. It may be the side length of the triangle, or it may correspond to a different length, such as the triangle’s altitude. The area formula depends on which correspondence the problem specifies.
This volume formula adds the areas of thin slices across the interval. In this tutorial, the main task is to express \(A(x)\) using the length of the corresponding chord in the base region.
Deriving the Area of an Equilateral Triangle
Let an equilateral triangle have side length \(s\). Draw its altitude from one vertex to the opposite side. The altitude bisects the opposite side, making two right triangles with hypotenuse \(s\) and one leg \(s/2\). By the Pythagorean theorem, the altitude \(h\) satisfies
Therefore,
The area of a triangle is one half its base times its perpendicular height. Using \(s\) as the base gives
Consequently, if the base-region chord at position \(x\) is the side of the cross-sectional triangle and has length \(d(x)\), then \(s=d(x)\), so
The chord length must be nonnegative on the interval being integrated. It is usually found by subtracting the lower boundary of the base region from the upper boundary for vertical chords, or the left boundary from the right boundary for horizontal chords.
Worked Example: Equilateral Slices Above a Semicircular Base
Worked Example: Equilateral Slices Above a Semicircular Base
The base is the semicircle \(x^2+y^2\leq 4\) with \(y\geq 0\). At each \(x\), a vertical chord of the base is the side of an equilateral-triangle cross-section. Find the volume.
Step 1: Find the chord length. The semicircle’s lower boundary is \(y=0\), and its upper boundary is \(y=\sqrt{4-x^2}\). Thus the vertical chord length is
The semicircle extends from \(x=-2\) to \(x=2\), so these are the integration bounds.
Step 2: Find the cross-sectional area. The chord is the triangle’s side, so
Step 3: Integrate the areas.
Check: The antiderivative \(4x-x^3/3\) differentiates to \(4-x^2\), as required. The volume is positive, and if lengths are measured in units, its units are cubic units. The result is \(\frac{8\sqrt{3}}{3}\) cubic units.
When the Chord Corresponds to Another Length
Not every problem says that the chord is a side. Sometimes it corresponds to a different length of the cross-section. The relevant fact about similar figures is that a length scale factor \(k\) multiplies every length by \(k\), and multiplies area by \(k^2\).
For equilateral triangles, this can be seen directly. If a reference equilateral triangle has side \(s_0\) and area \(A_0\), and a second triangle has every length \(k\) times as large, then its side is \(ks_0\). Its area is therefore
In particular, the altitude is \(h=(\sqrt{3}/2)s\). Solving for the side gives \(s=2h/\sqrt{3}\). Substituting this into the side-length area formula shows that if the chord is the altitude, then
Worked Example: The Chord Is the Triangle’s Altitude
Worked Example: The Chord Is the Triangle’s Altitude
The base region lies between \(y=0\) and \(y=1-x^2\) for \(-1\leq x\leq 1\). At each \(x\), the vertical chord is the altitude of an equilateral-triangle cross-section. Find the volume.
Step 1: Find the chord length. The upper boundary is \(1-x^2\) and the lower boundary is \(0\), so
On \([-1,1]\), this length is nonnegative and vanishes at both endpoints.
Step 2: Convert altitude to area. Because the chord is the altitude, not the side, use \(A=h^2/\sqrt{3}\):
Step 3: Integrate. Expand the square and evaluate:
Check: Differentiating \(x-2x^3/3+x^5/5\) gives \(1-2x^2+x^4=(1-x^2)^2\), the integrand before its constant factor. The volume is positive and is \(\frac{16\sqrt{3}}{45}\) cubic units.
Worked Example: A Fixed Similarity Ratio
Worked Example: A Fixed Similarity Ratio
The base region lies between \(y=x^2\) and \(y=1\) for \(-1\leq x\leq 1\). At each \(x\), the vertical chord is one-half of the side length of an equilateral-triangle cross-section. Find the volume.
Step 1: Find the base chord. The vertical chord has length
Since this is one-half of the triangle’s side length, the side length is \(s(x)=2d(x)=2(1-x^2)\).
Step 2: Use the scale factor. The side is twice the chord, so the linear scale factor from chord length to side length is \(2\). Equivalently, directly substitute the side length into the triangle area formula:
Step 3: Integrate.
Check: The antiderivative differentiates to \(1-2x^2+x^4\), and the factor \(\sqrt{3}\) is unchanged. The area at each slice is nonnegative on the stated interval, so the computed volume \(\frac{16\sqrt{3}}{15}\) cubic units is consistent with the geometry.
A Reliable Setup and a Common Pitfall
For each volume problem, identify the slicing direction and write the chord as the difference between the appropriate boundaries. Then determine which length of the triangle the chord represents. Only after that should you form the area and integrate over the interval where the base region exists.
Choose the variable and bounds that describe the base region along the slicing direction.
Subtract the two boundary values to obtain a nonnegative length \(d(x)\) or its horizontal-slice equivalent.
Use the side formula if the chord is a side; account for the fixed scale factor if it corresponds to another length.
Integrate the area over the base interval, then check the bounds, sign, and cubic units.
A frequent mistake is to use the side-length formula when the chord is actually the altitude. For instance, if the chord length is \(h\), the incorrect calculation \(A=(\sqrt{3}/4)h^2\) treats \(h\) as a side. The correct calculation first uses \(s=2h/\sqrt{3}\), giving
These two coefficients are different. The error changes every cross-sectional area, and therefore changes the volume. Likewise, if a chord is half the side, the side is twice the chord and the area is multiplied by \(2^2=4\), not merely by \(2\).
Units provide a useful check: a chord has units of length, its square has units of area, and integrating area with respect to a length gives units of volume. A negative chord length or a negative volume usually signals an incorrect subtraction order or bounds.
Check Your Understanding
For each question, identify the geometric length before using an area formula.
- Derive the altitude of an equilateral triangle with side length \(s\) using the Pythagorean theorem.
- A chord of length \(d\) is the side of an equilateral-triangle slice. Write its cross-sectional area.
- A chord of length \(h\) is the altitude of an equilateral-triangle slice. Write its area in terms of \(h\).
- If the side of each equilateral slice is three times the base chord \(d(x)\), what is the area as a function of \(d(x)\)?
- For the base between \(y=0\) and \(y=1-x^2\), \(-1\leq x\leq1\), write the volume integral when the chord is the triangle’s side. Do not evaluate it.
