Hexagonal pyramid cut by an inclined projecting plane
A competition entry for a pavilion uses a regular hexagonal pyramid as a roof. The base hexagon has side 30 mm at scale 1:100 and lies on the horizontal plane; the apex is 55 mm above the base centre. The model is to be cut by a plane P that is perpendicular to the vertical projection plane, makes 25° with the horizontal plane, and passes through a point on the axis 30 mm above the base. The jury wants a true-shape drawing of the cut face to calculate cladding. A student proposes to read the section directly from the plan, arguing that the plan already shows all six intersection points. The workshop needs to know whether the section is a hexagon or includes a piece of the base, and whether a cambio de plano or an abatimiento is the safer route. The base edges are oriented so that two sides are parallel to the vertical plane; this orientation is fixed.
1. Determine whether plane P cuts the base of the pyramid or only the six lateral faces. Justify the number of sides of the section.
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Use a vertical section through the axis. In that auxiliary view, P appears as a line rising at 25° and passing through the axis at height 30 mm. Where this line reaches the base plane (height 0), its horizontal distance from the axis is
d = 30 / tan 25° ≈ 64.3 mm.
The base hexagon has circumradius equal to its side, 30 mm, and apothem 30 cos 30° ≈ 25.98 mm. Since 64.3 mm exceeds 30 mm, P meets the base plane outside the hexagon. Therefore P does not cut the base; the section is bounded entirely by lateral faces.
Now check the six lateral edges. At the apex, height = 55 mm, which is above P at the axis (30 mm). At every base vertex, height = 0 mm, which is below P. Along each straight lateral edge the signed height difference relative to P changes continuously from positive at the apex to negative at the base, so each of the six edges meets P exactly once. Hence the section has six vertices, one on each lateral edge, and is a hexagon. Because no base edge is involved, the section remains a closed hexagon.
2. Compare a cambio de plano vertical with an abatimiento about the horizontal trace for obtaining the true shape. Which is safer here, and what are the key steps?
Read the worked answer
Plane P is perpendicular to the vertical projection plane, so its vertical projection is a straight line. Its horizontal trace h is the line where P meets the horizontal plane; that trace lies outside the base.
Cambio de plano vertical: choose a new vertical plane parallel to P. Its ground line must be parallel to h. Project the six section points from the plan perpendicular to the new ground line, then transfer each point’s height from the old vertical projection. Because the new vertical plane is parallel to P, the new vertical projection shows the section in true shape. This is the safer route: the heights are already known, and the operation is a direct projection rather than a rotation.
Abatimiento about h: rotate P about its horizontal trace into the horizontal plane. For each point, the coordinate along h is already true; the perpendicular distance from h in space must be recovered from the point’s height and the 25° slope. This is valid but more error-prone, because each point requires a separate height-to-true-distance conversion and the trace lies far from the base.
Recommended sequence: locate the six intersection points in plan and elevation; draw the new ground line parallel to h; project and transfer heights; connect the six true-shape vertices. Check that the resulting polygon is convex and that its side lengths match the corresponding lateral-face cuts.
3. If the hexagon were rotated so that a vertex, rather than a side, faced the vertical plane, would the section still be a hexagon? What would change?
Read the worked answer
Yes, the section would still be a hexagon. The argument depends only on the apex being above P and every base vertex being below P. Rotating the base about the vertical axis does not change those heights, so each of the six lateral edges still crosses P once. The number of sides is therefore unchanged.
What changes is the true shape: the horizontal distances from the axis to the base vertices in the direction of steepest slope change with the rotation, so the six intersection points lie at different heights and at different distances along P. The plan projection also changes, and the maximum width of the section may occur between a different pair of vertices.
The student’s claim that the plan already gives the true shape remains false in either orientation: the plan is an orthogonal projection onto the horizontal plane, and P is inclined at 25°, so distances measured up the slope are foreshortened. Only the direction parallel to the horizontal trace is true in plan.