When a plane intersects a right circular cylinder perpendicular to its bases, the resulting cross-section is a rectangle. And this fundamental geometric concept bridges the gap between three-dimensional solids and two-dimensional plane figures, serving as a cornerstone in calculus, engineering design, and architectural drafting. Understanding this interaction requires visualizing the spatial orientation of the cutting plane relative to the cylinder’s axis and generators Practical, not theoretical..
Understanding the Spatial Orientation
To fully grasp the geometry, we must first define the components involved. A right circular cylinder consists of two parallel, congruent circular bases connected by a lateral surface. In practice, the line segment joining the centers of the bases is the axis of the cylinder. The generators (or elements) are the infinite lines ruling the lateral surface, running parallel to the axis.
The bases lie in parallel planes. A plane that is perpendicular to the bases is, by definition, perpendicular to the planes containing the circles. In a right cylinder, the axis is perpendicular to the bases. Which means, a plane perpendicular to the bases is parallel to the axis of the cylinder.
Most guides skip this. Don't.
This parallel relationship between the cutting plane and the cylinder’s axis is the critical factor determining the shape of the intersection. Now, because the plane runs parallel to the generators, it slices through the lateral surface along two distinct, parallel lines. It cuts the bottom base along a chord (or diameter) and the top base along a congruent, parallel chord. The four intersection segments—two on the bases and two on the lateral surface—form a four-sided polygon with four right angles: a rectangle.
Quick note before moving on.
The Geometry of the Rectangular Cross-Section
The dimensions of this rectangle depend entirely on the position of the cutting plane relative to the cylinder’s central axis. Let the cylinder have a radius $r$ and a height $h$ (the perpendicular distance between bases).
Case 1: The Plane Contains the Axis (Axial Section)
If the cutting plane passes directly through the axis of the cylinder, it splits the solid into two symmetrical halves. The intersection with each base is a diameter of length $2r$. The intersection with the lateral surface consists of two generators (elements) running the full height $h$ of the cylinder.
- Width of rectangle: $2r$ (the diameter).
- Height of rectangle: $h$ (the height of the cylinder).
- Area: $A = 2rh$. This is the maximum possible rectangular cross-section obtainable from a given cylinder. It is often called the axial section or longitudinal section.
Case 2: The Plane is Parallel to the Axis but Offset (Parallel Section)
If the cutting plane is parallel to the axis but does not contain it, the intersection with the bases becomes chords rather than diameters. Let the perpendicular distance from the cylinder's axis to the cutting plane be $d$ (where $0 < d < r$) That's the part that actually makes a difference..
- Width of rectangle: The length of the chord. Using the Pythagorean theorem on the radius, the half-chord length is $\sqrt{r^2 - d^2}$. The full width is $w = 2\sqrt{r^2 - d^2}$.
- Height of rectangle: $h$ (unchanged, as the plane remains parallel to the axis).
- Area: $A = 2h\sqrt{r^2 - d^2}$. As $d$ increases from $0$ to $r$, the width decreases from $2r$ to $0$. When $d = r$, the plane is tangent to the cylinder, and the "rectangle" degenerates into a single line segment (a generator) of height $h$.
Mathematical Derivation Using Analytic Geometry
We can rigorously prove the rectangular nature of this cross-section using a 3D coordinate system. Here's the thing — place the cylinder such that its axis aligns with the $z$-axis. The equation of the cylinder is: $x^2 + y^2 = r^2$ The bases are the planes $z = 0$ and $z = h$ Worth keeping that in mind..
A plane perpendicular to the bases (the $xy$-plane) must have a normal vector with a zero $z$-component. Its general equation is: $ax + by = c \quad \text{(where } a^2 + b^2 \neq 0\text{)}$ This plane is vertical (parallel to the $z$-axis) Less friction, more output..
The intersection curve satisfies both equations simultaneously:
- $x^2 + y^2 = r^2$
- $ax + by = c$
Equation (2) represents a straight line in the $xy$-plane. The intersection of a line and a circle in 2D yields either two points (secant), one point (tangent), or no points. Here's the thing — equation (1) represents a circle. In 3D, because $z$ is unrestricted by the plane equation (it does not appear in $ax+by=c$), the $z$-coordinate is free to vary between $0$ and $h$.
So, for every valid $(x, y)$ pair solving the circle-line system, $z$ traces a vertical line segment from $0$ to $h$ The details matter here..
- If the line cuts the circle (secant), we get two distinct vertical lines $\rightarrow$ Rectangle.
- If the line is tangent, we get one vertical line $\rightarrow$ Line Segment (Degenerate Rectangle).
- If the line misses the circle, there is no intersection.
This algebraic approach
This algebraic approach confirms the geometric intuition developed earlier. Solving the system
[ \begin{cases} x^{2}+y^{2}=r^{2}\[2pt] ax+by=c \end{cases} ]
in the (xy)-plane is equivalent to finding the distance from the origin to the line (ax+by=c). The perpendicular distance is
[ d=\frac{|c|}{\sqrt{a^{2}+b^{2}}};, ]
which is precisely the offset of the cutting plane from the cylinder’s axis. Substituting (y = \frac{c-ax}{b}) (assuming (b\neq0); the case (b=0) is handled symmetrically) into the circle equation yields a quadratic in (x):
[ x^{2}+\left(\frac{c-ax}{b}\right)^{2}=r^{2} ;\Longrightarrow; \left(1+\frac{a^{2}}{b^{2}}\right)x^{2}-\frac{2ac}{b^{2}}x+\frac{c^{2}}{b^{2}}-r^{2}=0 . ]
Multiplying by (b^{2}) and simplifying gives
[ (a^{2}+b^{2})x^{2}-2acx+(c^{2}-r^{2}b^{2})=0 . ]
The discriminant of this quadratic is
[ \Delta = (2ac)^{2}-4(a^{2}+b^{2})(c^{2}-r^{2}b^{2}) =4b^{2}\bigl(r^{2}(a^{2}+b^{2})-c^{2}\bigr) . ]
Real solutions exist iff (\Delta\ge0), i.e. (|c|\le r\sqrt{a^{2}+b^{2}}), which is exactly the condition (d\le r).
[ x_{1,2}= \frac{ac\pm b\sqrt{r^{2}(a^{2}+b^{2})-c^{2}}}{a^{2}+b^{2}} . ]
The corresponding (y)-coordinates follow from (ax+by=c). The distance between the two intersection points in the (xy)-plane is
[ \sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}} = \frac{2b\sqrt{r^{2}(a^{2}+b^{2})-c^{2}}}{a^{2}+b^{2}} . ]
Using (d=|c|/\sqrt{a^{2}+b^{2}}) this simplifies to
[ \text{width}=2\sqrt{r^{2}-d^{2}} , ]
which matches the chord length derived geometrically in Case 2. Because the plane contains no (z)-term, the (z)-coordinate is unrestricted; each admissible ((x,y)) pair therefore generates a vertical segment from (z=0) to (z=h). The two vertical segments are parallel and separated by the width just computed, forming a rectangle of height (h) and area
[ A = h\cdot\bigl(2\sqrt{r^{2}-d^{2}}\bigr)=2h\sqrt{r^{2}-d^{2}} . ]
When (d=0) (the plane passes through the axis) the width attains its maximum (2r), reproducing the axial section of Case 1. As (d\to r) the width shrinks to zero and the rectangle collapses to a single generator, the degenerate case noted earlier.
Conclusion
The intersection of a right circular cylinder with any plane that is perpendicular to its bases is always a rectangle (or a line segment in the limiting tangent case). The rectangle’s height equals the cylinder’s height (h), while its width depends on the offset (d) of the cutting plane from the cylinder’s axis:
- Axial section ((d=0)): width (=2r), area (=2rh) – the largest possible rectangular cross‑section.
- Parallel offset section ((0<d<r)): width (=2\sqrt{r^{2}-d^{2}}), area (=2h\sqrt{r^{2}-d^{2}}).
- Tangent section ((d=r)): width (=0), the intersection reduces to a single vertical line (a generator).
These results follow both from elementary chord geometry and from a straightforward analytic‑geometry derivation, confirming that the family of rectangular cross‑sections spans continuously from the maximal axial rectangle down to the degenerate line as the cutting plane slides outward from the cylinder’s core. Practically speaking, understanding this behavior is useful in applications ranging from engineering design (e. On top of that, g. , determining maximal load‑bearing sections of cylindrical members) to computer graphics, where slicing cylinders yields predictable rectangular primitives for further processing Worth keeping that in mind..
