Cross Sections and Solids of Revolution

What a cut exposes

Slice a solid with a flat plane and look at the face you have opened. That flat region is the cross section.

Two things decide its shape: the solid, and the direction of the cut. The same cylinder gives a circle, a rectangle or an ellipse depending only on how the knife is held.

Slicing the standard solids

Solid	Cut parallel to the base	Cut through the axis	Slanted cut
Cylinder	circle	rectangle	ellipse
Cone	circle	triangle	ellipse, parabola or hyperbola
Sphere	circle	circle	circle
Cube	square	square or rectangle	triangle, hexagon and more
Square pyramid	square	triangle	trapezoid

Two rows repay a second look.

The cone’s slanted cuts produce the curves called conic sections, which is where that name comes from. A shallow cut gives an ellipse, a cut parallel to the side gives a parabola, and a steep cut through both halves gives a hyperbola.

The sphere’s row never changes. Every plane that meets a sphere meets it in a circle, whatever its angle.

Why every flat slice of a sphere is a circle

The sphere is the only row above with one answer, and the reason is the definition.

A sphere of radius $r$ is the set of points exactly $r$ from its center $O$.

Cut it with any plane. Drop a perpendicular from $O$ to that plane and call the foot $F$, at distance $d$. Take any point $P$ where the plane meets the sphere.

Triangle $OFP$ has a right angle at $F$, so

$$d^2 + (FP)^2 = r^2 \qquad\Longrightarrow\qquad FP = \sqrt{r^2 - d^2}$$

The right-hand side contains $r$ and $d$ and nothing about $P$. Every point of the slice is the same distance from $F$, and a set of points at a fixed distance from a fixed point in a plane is a circle.

$$\text{slice radius} = \sqrt{r^2 - d^2}$$

The cut through the center has $d = 0$ and gives radius $r$, the largest possible slice — the great circle. As the plane slides outward the radius shrinks, reaching zero when $d = r$ and the plane touches at a single point.

Spinning a flat shape

The same solids arrive a second way. Take a two-dimensional shape, pick a line, and spin the shape about that line. The region it sweeps through is a solid of revolution.

A rectangle spun about one of its own sides sweeps out a cylinder. The side on the axis becomes the cylinder’s axis, and the side opposite becomes its curved surface.

A right triangle spun about one of its legs sweeps out a cone. The leg on the axis is the height, the other leg is the base radius, and the hypotenuse traces the slanted surface.

A semicircle spun about its diameter sweeps out a sphere.

Flat shape	Axis	Solid produced
Rectangle	one of its sides	cylinder
Rectangle	a line beside it	tube with a hollow core
Right triangle	one of its legs	cone
Right triangle	its hypotenuse	two cones joined base to base
Semicircle	its diameter	sphere
Circle	a line outside it	torus, the shape of a doughnut

Reading a solid backwards

Given a solid, the question flips: which flat shape and which axis produce it?

Look at the solid’s cross section through its axis, then keep half of it.

Solid	Half of the axial cross section	Axis
Cylinder, radius $4$, height $9$	a $4 \times 9$ rectangle	the side of length $9$
Cone, radius $5$, height $12$	a right triangle with legs $5$ and $12$	the leg of length $12$
Sphere, radius $7$	a semicircle of radius $7$	its diameter

The axial cross section of a cylinder is a full rectangle $8$ wide, and only half of it is spun — the other half is where the first half lands.

Slicing a cube

A cube rewards experiment more than any other solid.

Cut	Cross section
Parallel to a face	square
Parallel to an edge, tilted	rectangle
Through three edges at one corner	triangle
Perpendicular to a long diagonal, through the center	regular hexagon

The hexagon is the surprise. That plane meets all six faces, and a plane meeting a face in a straight segment produces one side per face.

Worked examples

Common mistakes

Practice problems