Why Cross-Sections Can Determine Volume
A solid can lean, twist in appearance, or have cross-sections whose shapes change with height. These features may make its volume hard to judge from its outside boundaries alone. But if we know the area of every horizontal cross-section, we can determine the volume by adding the contributions from all heights.
This observation leads to the Cavalieri principle. It allows us to compare two solids without requiring their cross-sections to have the same shape. The essential requirement is that the areas match at corresponding heights throughout the same vertical interval.
The planes must be parallel to one another, and the heights must correspond: a section at height \(z\) in one solid is compared with a section at that same height in the other. The cross-sections themselves need not look alike. One could be triangular and the other rectangular, for example.
Why the Principle Works
For a solid whose cross-sectional area at height \(z\) is \(A(z)\), the cross-section volume formula gives its volume as the integral of area over height:
This formula is understood to apply when the area function is integrable over the interval. Its basic idea is to divide the solid into thin horizontal layers. A layer of thickness \(\Delta z\) and area approximately \(A(z)\) contributes approximately \(A(z)\Delta z\) to the volume. Adding these contributions and taking the limit gives the integral.
Here is the complete deduction of the comparison rule. For two solids over the same interval, apply the volume formula to each:
If their cross-sectional areas agree at every height, then the integrands agree throughout the interval. Therefore their integrals agree:
The principle is not claiming that the solids have the same shape. It says that their volumes agree because each matching layer contributes the same amount of volume.
A Shear Preserves the Areas of Horizontal Sections
A useful way to build a different solid is to shift each horizontal cross-section sideways by an amount that depends on height. This changes where the sections sit but does not change their areas. For instance, the transformation
leaves height unchanged and shifts the section at height \(z\) horizontally by \(2z/5\). Within any fixed horizontal plane, every point is translated by the same amount. Translation preserves area, so corresponding horizontal sections before and after the transformation have equal areas.
Worked Example: A Triangular Prism and Its Sheared Copy
Take a triangular base with vertices \((0,0)\), \((4,0)\), and \((0,3)\), and extend it vertically from \(z=0\) to \(z=5\). Compare this straight prism with the solid obtained by shifting each section according to \(x'=x+2z/5\), \(y'=y\), and \(z'=z\).
First find the area of the triangular section. Its base is \(4\) and its perpendicular height is \(3\), so
Every horizontal section of the straight prism has area \(6\). At a fixed height \(z\), the transformed section is a translation of that triangle by \(2z/5\), so it also has area \(6\). Both solids occupy the same height interval, \([0,5]\). Thus
Check the integral calculation: the derivative of \(6z\) is \(6\), and \(6(5)-6(0)=30\). The shift is \(0\) at the bottom and \(2\) at the top, so the second prism leans while its horizontal sections retain area \(6\). Both volumes are \(30\) cubic units.
Matching Areas Do Not Require Matching Shapes
Cavalieri's principle applies even when two solids have different cross-sectional shapes at every height. What matters is the area of each section, not its perimeter, angles, or orientation within the plane.
Worked Example: Square and Rectangular Sections
Let solid \(S_1\) have square horizontal sections at heights \(0\le z\le2\), with side length \(2-z\). Let solid \(S_2\) have rectangular sections at those heights, with side lengths \((2-z)/2\) and \(2(2-z)\).
At each height, the square section has area
The rectangular section has area
The sections have equal areas throughout the shared interval, so the principle says the volumes agree. Compute either volume by expanding the area function:
For a check, differentiating the antiderivative gives \(z^2-4z+4=(2-z)^2\), the section area. The square sections begin with side length \(2\), while the rectangular sections begin with side lengths \(1\) and \(4\); nevertheless, the equal area at each height gives each solid volume \(8/3\) cubic units.
Using Piecewise Cross-Section Areas
The area functions need not have one formula over the whole height interval. If the section-area rule changes, calculate the volume on each subinterval and add the results. For a comparison, it is enough that the two solids have the same area function on each corresponding part.
Worked Example: Piecewise Areas with Different Section Shapes
Suppose solid \(S_1\) has rectangular sections of width \(1\) and length \(A(z)\), while solid \(S_2\) has triangular sections with base \(2\) and perpendicular height \(A(z)\). Let the shared area function be
The rectangle has area \(1\cdot A(z)=A(z)\). The triangle has area \(\frac12(2)A(z)=A(z)\). Thus the areas match at every height, and Cavalieri's principle applies. Calculate the common volume by integrating over the two intervals:
Check each antiderivative: \(2z+z^2/2\) differentiates to \(2+z\), and \(4z-z^2/2\) differentiates to \(4-z\). The first interval contributes \(5/2\), the second contributes \(4\), and their sum is \(13/2\) cubic units. The solids have different section shapes but the same volume.
Conditions and a Common Pitfall
Before applying the principle, check that the solids have the same height interval and that the compared sections are cut by planes with the same orientation at matching heights. Then verify equality of the section areas over the entire interval, not just at a few selected heights. If either solid has an integrable area function, the volume formula applies; in these comparisons, the matching area function gives the same integral for both.
A common error is to use a slanted edge as though it were the vertical height. Consider a \(4\)-by-\(3\) rectangular prism of vertical height \(5\), sheared so that its top shifts horizontally by \(2\). Each horizontal section still has area \(12\), and the height interval is still \([0,5]\). The correct calculation is
A slanted side edge has length \(\sqrt{5^2+2^2}=\sqrt{29}\). The incorrect calculation \(12\sqrt{29}\) treats that edge length as the height, even though it is not the distance between the bottom and top horizontal planes. The section integral uses vertical height, so the shear does not alter the volume.
Check Your Understanding
Use the conditions of the Cavalieri principle, not visual similarity alone, to answer these questions.
- Two bounded solids occupy the same height interval. What must be true of their horizontal cross-sectional areas for Cavalieri's principle to guarantee equal volumes?
- A solid's horizontal sections are translated sideways by a different amount at each height. Why can their areas remain unchanged?
- A square section has side length \(3\). A rectangle has side lengths \(1\) and \(9\). Do these sections have the same area, and what does this tell you about the importance of section shape?
- Why is it not valid to replace the vertical height of a sheared solid with the length of a slanted side edge?
- If the cross-sectional area changes formula at an intermediate height, how should the volume be calculated?
