Cross-Sections of 3D Figures Math Example 2

Solution

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   Step 1: (a) Parallel to a face: The plane slices through the cube parallel to one of its square faces. Every point on the cut is equidistant from two opposite faces, producing a square cross-section congruent to the face.
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   Step 2: (b) Diagonal cut through four edges: If the plane cuts through the midpoints of four parallel edges and is perpendicular to those edges' face, it produces a rectangle. If it cuts corner-to-corner across opposite edges, it produces an oblong rectangle (or specifically a rectangle with width equal to the cube's side and length equal to the face diagonal $s\sqrt{2}$).
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   Step 3: (c) Through three vertices not on the same face: Choosing three vertices that are each on a different pair of opposite faces yields an equilateral triangle. For a unit cube, vertices like $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ define an equilateral triangle cross-section.
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   Step 4: In general, cross-sections of a cube can be triangles, rectangles, squares, pentagons, or hexagons depending on the cutting plane's orientation.