Cube Section True Shape and Template
A solid cube of side 80 mm is to be sectioned for a museum display model. The cube is labelled so that the bottom face is ABCD in order and the top face EFGH is directly above A, B, C, D respectively. The cutting plane passes through the midpoints of three edges: AB on the bottom front, BC on the bottom right, and CG on the vertical right edge. The model maker needs the true shape of the section face, its area, and a check that the section is a regular hexagon before machining a template. The workshop has only a 2D projector and wants to mark the template from the front view, because the front view already shows the cut as a six-sided outline. The designer must confirm which cube edges the plane actually cuts and whether the projected outline can be used directly for the template.
1. Determine which edges of the cube are cut by the plane and identify the section polygon. Show how you know no other edges are cut.
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Use coordinates: A = (0,0,0), B = (80,0,0), C = (80,80,0), D = (0,80,0), E = (0,0,80), F = (80,0,80), G = (80,80,80), H = (0,80,80). The three midpoints are P = (40,0,0), Q = (80,40,0), R = (80,80,40). Vectors PQ = (40,40,0) and PR = (40,80,40), so a normal is PQ × PR = (1600, −1600, 1600), proportional to (1, −1, 1). The plane is x − y + z = 40. Checking each edge for a parameter in [0,1] gives intersections at P on AB, Q on BC, R on CG, S = (40,80,80) on GH, T = (0,40,80) on HE, and U = (0,0,40) on EA. No other edge meets the plane within the cube. The section is hexagon P-Q-R-S-T-U in that order around the cut.
2. Calculate the true lengths of the section edges and the area of the section. State whether the section is regular.
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Sides: PQ = √(40² + 40²) = 56.57 mm; QR = √(0² + 40² + 40²) = 56.57 mm; RS = √(40² + 0² + 40²) = 56.57 mm; ST = √(40² + 40² + 0²) = 56.57 mm; TU = √(40² + 0² + 40²) = 56.57 mm; UP = √(0² + 40² + 40²) = 56.57 mm. All six sides are equal. The plane normal (1, −1, 1) is parallel to a body diagonal of the cube, so the section is perpendicular to a threefold axis and is a regular hexagon. Area = (3√3/2)s² = 2.598 × 3200 = 8314 mm², or 8.31 × 10³ mm².
3. The model maker proposes to machine the template by marking the section directly on a 2D projection using the front-view lengths. Evaluate this proposal.
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The front view projects the hexagon onto the x-z plane. PQ projects to a horizontal length of 40 mm, but its true length is 56.57 mm; QR projects to a vertical length of 40 mm, also foreshortened. Marking directly from the front view would produce a template with some edges only 40 mm instead of 56.57 mm, a 29% error, and the hexagon would not close correctly. The true shape must be obtained by an auxiliary view with the line of sight perpendicular to the cutting plane, or by coordinates. The 2D projector can be used only after the true-shape view is constructed, for example by projecting the six cut points onto an auxiliary plane parallel to the section.