9.3 Cross Sections of 3D Figures & Solids of Revolution

9.3 Cross Sections of 3D Figures & Solids of Revolution

Geometry of Planar Cross Sections

A planar cross section is the two-dimensional geometric region formed by the intersection of a three-dimensional solid with a cutting plane. Cross-sectional analysis connects spatial geometry to multi-variable calculus, computer-aided design, and medical imaging (tomography).

The fundamental topological and geometric constraints governing cross sections include:

1. Dimensionality Reduction: Slicing an $n$-dimensional Euclidean object with an $(n-1)$-dimensional flat hyperplane produces an $(n-1)$-dimensional subset. In $\mathbb{R}^3$, the intersection of a solid and a 2D plane is always a 2D planar figure (or degenerate 1D line segment/0D point if tangent).
2. Convex Polyhedron Face Constraint: If a convex polyhedron has $F$ planar faces, any planar cross section can intersect at most $F$ faces. Furthermore, because both the polyhedron faces and the cutting plane are flat, the intersection of the cutting plane with any single face is at most a single straight line segment. Therefore, the cross section of a convex polyhedron is always a convex polygon whose number of edges $E_{\text{slice}}$ satisfies:

$$3 \le E_{\text{slice}} \le F$$

---

Cross Sections of Polyhedra: The Cube and Tetrahedron

Understanding how cutting planes intersect standard polyhedra is a benchmark skill on the FTCE Mathematics 6-12 examination:

• Slicing a Solid Cube ($F = 6$):
  • Triangles ($E_{\text{slice}} = 3$): Formed when the cutting plane slices through three mutually adjacent faces meeting at a single corner. Slicing equidistant from the vertex produces an equilateral triangle; varying distances produce isosceles or scalene triangles.
  • Quadrilaterals ($E_{\text{slice}} = 4$): Slicing parallel to any cube face yields a square. Slicing obliquely through four parallel lateral faces yields a non-square rectangle, a rhombus, or a general parallelogram. Slicing through opposite faces at non-parallel angles yields an isosceles trapezoid or general trapezoid.
  • Pentagons ($E_{\text{slice}} = 5$): Formed when the cutting plane intersects exactly 5 faces of the cube, clipping five edges without intersecting the sixth face.
  • Hexagons ($E_{\text{slice}} = 6$): Formed when the plane intersects all 6 faces of the cube. If the plane passes through the midpoints of six consecutive edges forming a non-coplanar closed cycle and passes through the cube's geometric center, the cross section is a perfect regular hexagon!
  • Heptagons ($E_{\text{slice}} \ge 7$): Strictly impossible. Because a cube has exactly 6 faces and is convex, a flat plane cannot generate 7 or more edges.
• Regular Tetrahedron ($F = 4$): Can only produce triangles ($E_{\text{slice}} = 3$) or quadrilaterals ($E_{\text{slice}} = 4$). Remarkably, a plane passing symmetrically through four edges parallel to two opposite skew edges forms a square!

---

Cross Sections of Curved Solids: Cylinders, Cones, and Spheres

• Right Circular Cylinder:
  • Parallel to base: Circle of radius $r$.
  • Oblique plane cutting through the lateral surface without intersecting bases: Ellipse.
  • Perpendicular to base (vertical cut through axis): Rectangle of dimensions $2r \times h$ (a square if $h = 2r$).
  • Plane cutting through base and lateral wall: Truncated parabolic or hyperbolic segment.
• Right Circular Cone and Conic Sections:
  • Parallel to base: Circle.
  • Tilted cutting plane through one nappe at an angle greater than the generator slant: Ellipse.
  • Plane parallel to exactly one generating line (element/slant edge) of the cone: Parabola.
  • Plane parallel to the cone's central vertical axis (or cutting both nappes of a double cone at an angle shallower than the generator): Hyperbola.
  • Degenerate cross sections passing through the apex: A single point (tangent to apex), a single straight line (tangent along a generator), or two intersecting lines (vertical plane passing through the apex, which for a single bounded cone appears as an isosceles triangle).
• Sphere:
  • Every non-empty planar cross section of a sphere of radius $R$ is ALWAYS a circle.
  • If the cutting plane passes through the sphere's center (distance $d = 0$), the cross section is a great circle of maximum radius $r = R$.
  • If the plane is at perpendicular distance $d$ from the center ($0 < d < R$), the cross section is a small circle with radius:

$$r = \sqrt{R^2 - d^2}$$

• If $d = R$, the intersection is a single degenerate tangent point.

---

Solids of Revolution: Rotational Generation Mappings

A solid of revolution is a three-dimensional figure generated by revolving a two-dimensional planar region $360^\circ$ ($2\pi$ radians) around an axis of coplanar rotation.

• Standard Geometric Profiles:
  1. Rectangle: Rotating a rectangle of dimensions $r \times h$ around an axis containing one of its sides generates a right circular cylinder of radius $r$ and height $h$.
  2. Right Triangle: Rotating a right triangle with legs $r$ and $h$ around the leg of length $h$ generates a right circular cone of base radius $r$, altitude $h$, and slant height $l = \sqrt{r^2 + h^2}$.
  3. Semicircle: Rotating a semicircle of radius $r$ around its bounding diameter generates a full sphere of radius $r$.
  4. Right Trapezoid: Rotating a right trapezoid with parallel bases $r$ and $R$ around the leg perpendicular to the bases generates a conical frustum with base radii $r$ and $R$.
  5. Equilateral or Isosceles Triangle: Rotating around its altitude of symmetry yields a single cone. Rotating around its base yields a bicone (two identical cones joined base-to-base).

---

Quantitative Mensuration of Revolution Solids: Disk, Washer & Shell Methods

Calculus links cross-sectional geometry to rotational volume via single-variable integration:

• Disk Method (Solid of Revolution without Hole): When the planar region bounded by $y = f(x) \ge 0$, $y = 0$, $x = a$, and $x = b$ is revolved about the x-axis, each vertical strip of width $dx$ sweeps out a circular disk of radius $R(x) = f(x)$ and thickness $dx$:

$$V = \pi \int_a^b [f(x)]^2 \, dx$$

• Washer Method (Solid of Revolution with Central Void): When the region bounded between outer curve $y = R(x)$ and inner curve $y = r(x)$ is revolved about the x-axis, each vertical strip sweeps out an annular washer with outer radius $R(x)$ and inner radius $r(x)$:

$$V = \pi \int_a^b \left( [R(x)]^2 - [r(x)]^2 \right) dx$$

• Cylindrical Shell Method: Revolving a region bounded by $y = f(x)$ about the vertical y-axis decomposes the solid into concentric cylindrical shells of radius $x$, height $f(x)$, and thickness $dx$:

$$V = 2\pi \int_a^b x f(x) \, dx$$

---

Pappus's Centroid Theorems for Rotational Mensuration

Pappus of Alexandria established two powerful theorems uniting rotational mensuration with planar centroids:

Pappus's First Theorem (Surface Area): The area $A$ of a surface of revolution generated by revolving a plane curve $C$ about a non-intersecting coplanar axis equals the product of the arc length $L$ of $C$ and the distance traveled by the centroid of $C$:

$$A = 2\pi \bar{y} L$$

where $\bar{y}$ is the perpendicular distance from the curve's centroid to the axis of rotation.

Pappus's Second Theorem (Volume): The volume $V$ of a solid of revolution generated by revolving a plane region $R$ about a non-intersecting coplanar axis equals the product of the area $A$ of $R$ and the distance traveled by the centroid of $R$:

$$V = 2\pi \bar{y} A$$

where $\bar{y}$ is the perpendicular distance from the region's centroid to the axis of rotation.

---

Worked Exemplar: Solid of Revolution Generated by a Trapezoid

Problem: A rectangular region with vertices at $(2, 0)$, $(5, 0)$, $(5, 4)$, and $(2, 4)$ is revolved $360^\circ$ around the y-axis. Calculate the volume of the resulting three-dimensional solid using both the washer method and Pappus's Second Theorem.

• Solution via Washer Method: The solid generated is a hollow circular cylinder (thick pipe) with outer radius $R = 5$, inner radius $r = 2$, and height $h = 4$. The horizontal cross section at any height $y \in [0, 4]$ is a washer with area:

$$A_{\text{washer}} = \pi R^2 - \pi r^2 = \pi(5^2 - 2^2) = \pi(25 - 4) = 21\pi$$

Integrating across height $h = 4$:

$$V = \int_0^4 21\pi \, dy = 21\pi \times 4 = 84\pi$$

• Solution via Pappus's Second Theorem:

1. Area of the rectangle: width is $5 - 2 = 3$, height is $4 - 0 = 4$. Area $A = 3 \times 4 = 12$.
2. Centroid of the rectangle: By symmetry, the x-coordinate of the centroid is the midpoint of the horizontal span:

$$\bar{x} = \frac{2 + 5}{2} = 3.5$$

1. Distance traveled by the centroid during a $360^\circ$ rotation about the y-axis:

$$d = 2\pi \bar{x} = 2\pi(3.5) = 7\pi$$

1. Volume by Pappus:

$$V = 2\pi \bar{x} A = (7\pi)(12) = 84\pi$$

Both methods independently verify the identical volume of $84\pi$.

---

Cross-Section Slicing and Solid of Revolution Mapping Tables

Planar Cross-Section Slices of Common 3D Figures

Solids of Revolution Rotation Mappings