Cutting a box’s corner does not necessarily create a square edge. Even a cube can be sliced into a triangular or hexagonal section. How do we find the actual shape from a flat drawing? We will construct the section face by face rather than join arbitrary points directly.
What are we intersecting?
Prerequisites: quadrilaterals, lines and planes, projections.
A polyhedron is a solid whose surface consists of finitely many planar polygons. These polygons are faces; their shared sides are edges; edge endpoints are vertices. A cube has 6 faces, 12 edges and 8 vertices; count each shared edge once.
Here we consider convex polyhedra. The section of a solid by a plane consists of all points belonging to both solid and plane. When the plane passes through the interior, the section is a filled convex polygon: its sides lie in faces and its vertices on edges, sometimes at solid vertices. It is not just an outline: the outline bounds the section. A plane touching only a vertex or edge gives a degenerate intersection, a point or segment.
Projection differs: it can include images of points throughout the volume. A section moves nothing: it retains only points in the cutting plane itself. A cube’s screen silhouette does not determine its section’s shape.
Our cube and three given points
Take cube ABCDA₁B₁C₁D₁ with edge 5 m. Its bottom face is ABCD; A₁ is above A, B₁ above B, and so on. On vertical edges choose K∈AA₁, L∈BB₁, M∈CC₁ with AK=1 m, BL=2 m, CM=4 m. These three noncollinear points determine one plane.
1 · Two adjacent faces
2 · Through K parallel to LM
3 · Closed section KLMN
- Front and right faces. K and L share face ABB₁A₁, so KL is its intersection with the cutting plane inside the cube. Similarly, LM lies in face BCC₁B₁. These are adjacent section sides.
- The fourth point. The left and right cube faces are parallel. Their intersections with the same cutting plane are parallel: through K in face ADD₁A₁ draw a line parallel to LM. Call its intersection with DD₁ point N.
- Closure. Join N to M in the back face DCC₁D₁. We obtain KLMN. Since KL∥NM and LM∥KN, it is a parallelogram.
On the right face, going from L to M raises height by 4−2=2 m over one horizontal cube side. On the left face, parallel KN gives the same rise: DN=1+2=3 m. This lies within the 5 m edge. The section does not meet top or bottom faces: on each vertical face we have already constructed the entire intersection line, whose endpoints lie inside the vertical edges.
A drawn parallelogram need not be a spatial rectangle. In the front face, KL has horizontal run 5 m and rise 1 m; in the right face LM has run 5 m and rise 2 m. Their spatial lengths satisfy:
(KL)^(2) = (5)^(2) + (1)^(2) = 26, (LM)^(2) = (5)^(2) + (2)^(2) = 29.
Thus equal cube edges do not even make adjacent section sides equal. Measuring drawn lengths cannot replace Pythagoras.
Additional constructionThe main section has already been constructed using parallel faces.
Branch · The trace method
A plane’s trace on another plane is their intersection line, when they are distinct and not parallel. In our example, extend KL to meet line AB at T and LM to meet line BC at U. Both points lie outside the cube: T beyond A, U beyond B in the direction opposite C. Line TU is the cutting plane’s trace on base plane ABCD.
Let W=TU∩AD, also on the extension beyond A. W and K belong to both cutting plane and left face plane. Hence WK is their intersection line; WK∩DD₁ again gives N. Auxiliary lines may extend outside the solid, but only segments inside faces are section sides. If required lines are parallel and never meet, do not invent an intersection: choose another auxiliary face or use parallelism.
Checking the result
Adjacent section vertices must lie in a shared face. A shared edge of two faces must contain the same point, not two nearby points. The outline closes; in our non-degenerate case no side “floats” in space. Dashes indicate cube edges hidden from the observer or auxiliary lines as specified in the caption, not a different kind of geometry.
A cube has six faces, and each can contribute at most one side to a convex non-degenerate section. Thus polygons from triangles to hexagons are possible; seven sides are impossible. Do not transfer this restriction to nonconvex solids without checking.
Sources and limits
Face, trace and parallel auxiliary-line methods were checked against Ister’s grade-10 geometry, §§2 and 7. University of Luxembourg’s Cut the Cube confirms convexity and possible shapes; its classification using strips offers another view of a section as intersecting constraints. Numbers, notation and SVG are our own. This is a geometric model without material thickness or tool-use instructions.