Cross-Section
Geometry & MeasurementA cross-section is the 2-D shape obtained by cutting through a 3-D solid with a plane.
Definition
A cross-section is the 2-D shape you see when you slice straight through a 3-D object. Like cutting a loaf of bread: the slice face you see is the cross-section.
Example
Slice a cylinder horizontally and you see a circle. Slice it vertically along the center and you see a rectangle. Slice an orange through the middle and you see a circle of fruit.
Key Insight
The shape of a cross-section depends on the angle of the cut. Slicing a cone at different angles creates the four conic sections: circle, ellipse, parabola, and hyperbola. This is one of the most famous ideas in all of geometry.
Definition
A cross-section is the intersection of a 3-D solid with a plane, producing a 2-D figure. For a prism or cylinder, horizontal cross-sections are congruent to the base. Oblique cuts produce different shapes. Cross-sections are used in Cavalieri's principle: if every cross-section of two solids at the same height has equal area, the solids have equal volume.
Example
A cube cut by a diagonal plane through $4$ vertices produces a rectangular cross-section. Cut through $6$ edge midpoints at an angle and you get a regular hexagon. A sphere cut by any plane gives a circle (great circle if through center).
Key Insight
Cavalieri's principle uses cross-sections to compare volumes without direct computation. Archimedes used this method to find the volume of a sphere, $2000$ years before calculus, by comparing its cross-sections to those of known solids.
Definition
A cross-section of a solid $S$ is the set $S \cap H$ for a hyperplane $H$. For a solid of revolution generated by $f(x) > 0$ rotated about the x-axis, the cross-section at position $x$ is a disk of radius $f(x)$ and area $\pi f(x)^2$. Volume by the disk method: $V = \int \pi f(x)^2 \, dx$. The washer method handles solids with holes.
Example
For a solid of revolution from $y = \sqrt{x}$ on $[0,4]$ rotated about the x-axis: each cross-section at $x$ is a disk of radius $\sqrt{x}$, area $\pi x$. $V = \int_0^4 \pi x \, dx = \pi\left[\frac{x^2}{2}\right]_0^4 = 8\pi$.
Key Insight
The disk and shell methods for volumes of revolution are two equivalent ways to decompose a solid into infinitely thin cross-sections or shells. The choice between them depends on which integral is easier to compute, a recurring theme in applied calculus.