Sectioning a Hexagonal Pyramid for a Template
A hexagonal pyramid with base side 30 mm and axis 70 mm stands on its base on HP. A cutting plane is inclined at 30° to HP and passes through a point on the axis 40 mm above the base. In the front view the cutting line is drawn through that axis point, and the student shades the cut face and labels it “regular hexagon, 60 mm across corners.” The true shape is required for a sheet-metal template, and the lateral surface must be developed. The student has not drawn an auxiliary plane and has measured the section edges directly from the front view. The workshop needs the template to fit the actual pyramid, not an assumed regular section. The examiner asks whether the plane cuts all six lateral edges and how the true lengths of the section edges can be obtained.
1. Determine whether the cutting plane intersects all six lateral edges and what section polygon results. Justify with a calculation.
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Take the base centre as origin, with the axis vertical and the cutting plane rising in the +x direction. Its equation is z = 40 + x tan 30°. The base circumradius is 30 mm, so the leftmost base vertex has x = −30 mm. At that vertex the plane is at z = 40 − 30/√3 = 22.68 mm, which is above the base. At every other base vertex the plane is even higher. At the apex, x = 0 and z = 70 mm, while the plane is at z = 40 mm, so the apex is above the plane. Therefore the plane crosses every lateral edge between the base and the apex. The section has six vertices and is a hexagon. It is not a regular hexagon because the cutting plane is not parallel to the base. The fraction of each edge measured from the apex is t = 30/(70 + x/√3), which varies from about 0.344 to 0.570 around the pyramid, so the section sides are unequal. The student’s 60 mm across-corners value is an assumption, not a construction.
2. Describe how to obtain the true shape of the section and why the front view is not the true shape.
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The cutting plane is perpendicular to VP, so its front view is an edge view: the section appears as a straight line, not as a polygon. To obtain the true shape, draw an auxiliary reference line parallel to the cutting line in the front view and project the section points from the top view perpendicular to that line. Transfer the corresponding distances from the front view to the auxiliary view. The resulting figure is the true shape because the auxiliary plane is parallel to the cutting plane. Measuring the section edges directly in the front view is meaningless: those lengths are projected onto the edge view and are foreshortened or collapsed. The top view gives the plan positions of the section points, but it also does not show true lengths because the cutting plane is inclined to HP. Only the auxiliary view parallel to the cutting plane gives the template shape.
3. Explain how the section points are transferred to the lateral-surface development.
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First develop the whole lateral surface as six congruent isosceles triangles. The true slant edge is √(70² + 30²) = 76.16 mm, and each base side is 30 mm. Set out the six triangles on a development line using those true lengths. Each section point lies on a lateral edge, but its distance from the apex is foreshortened in the front view. Obtain the true distance from the apex to each section point from the auxiliary view or by rotating that edge into a true-length position. Step off those true distances from the apex along the corresponding developed edges. Join the six plotted points in order to form the section curve on the development. Do not transfer the front-view distances directly, and do not draw the development as if the section were a straight line across the triangles. The resulting template will then match the actual oblique cut.