Sections of Solids and True Shape of the Section

Syllabus limits: solids with the axis in vertical position only, and problems in which the true shape is given are excluded. So you will always be given the cutting plane and asked for the section — never the reverse.

Why sections exist

Module 1 said that hidden lines should not be dimensioned. A section is the answer: cut the solid with an imaginary plane, remove the near part, and draw what remains. Interior features that were hidden become visible outlines and can be dimensioned properly.

Terms

• Section plane (cutting plane) — the imaginary plane that cuts. Shown in the other view as a thin chain line thick at the ends, with arrows giving the direction of viewing.
• Section — the figure actually cut, the face exposed.
• Sectional view — the remaining solid, drawn with the section hatched.
• True shape of the section — the section seen perpendicular to the cutting plane, that is its real size and shape.

Hatching convention: thin continuous lines at 45°, evenly spaced, over the cut face only. Do not hatch the parts of the solid the plane passed through without cutting.

The critical distinction

The sectional view is not the true shape.

The sectional view shows the section as it appears in the standard front or top view, where it is foreshortened because the cutting plane is inclined to that plane of projection. The true shape requires a new direction of viewing, perpendicular to the cutting plane — an auxiliary projection.

Confusing them is the central error in this topic, and both are usually asked for in the same question.

Procedure — solid with vertical axis, cut by an inclined plane

Take a solid resting on the HP with the axis vertical, cut by a plane perpendicular to the VP and inclined to the HP. This is the standard case.

1. Draw the uncut solid in both views, exactly as in Module 2.
2. Draw the cutting plane in the FRONT view, as a line at the given angle to XY. Because the plane is perpendicular to the VP, it appears as a straight line in the front view — this is why the case is chosen.
3. Mark every intersection of that line with an edge or generator of the front view. Label them 1′, 2′, 3′, …
4. Project each point down to the corresponding edge in the top view, giving 1, 2, 3, …. Join them to obtain the section in the top view — the sectional top view.
5. Hatch the cut face.

For a cone or cylinder there are no edges to intersect, so first draw a set of generators — typically twelve, evenly spaced — in both views, and use their intersections as the points. The more generators, the smoother the curve.

Obtaining the true shape

The true shape is found by projecting perpendicular to the cutting plane.

6. Draw a new reference line $X_1Y_1$ parallel to the cutting plane line in the front view, placed clear of the drawing.
7. From each point 1′, 2′, 3′ … draw a projector perpendicular to $X_1Y_1$ — that is, perpendicular to the cutting plane.
8. On each projector, lay off the distance of the corresponding point from XY, measured in the top view.
9. Join the points. That figure is the true shape of the section.

Why step 8 works. Rotating the section about the cutting plane's line does not change any point's distance from the VP, and that distance appears in the top view as the distance from XY. So the top view supplies the one dimension the front view cannot.

This is the same conservation principle as the change-of-position method of Module 2 — tilt about an axis and distances measured along that axis are preserved — appearing for the third time in the course. Recognising it as one idea rather than three separate recipes is what makes the module manageable.

The sections you should be able to predict

Being able to name the expected shape lets you check a construction at a glance.

Cone — the conic sections:

Cylinder — perpendicular to axis: circle; inclined: ellipse; parallel to axis: rectangle.

Prism — a plane cutting all lateral faces gives a polygon with as many sides as faces cut.

Pyramid — similar, and a plane through the apex gives a triangle.

The check worth running every time: count the edges the cutting line crosses in the front view. The true shape has exactly that many sides (or is a smooth curve for a cone or cylinder). If your true shape has five sides but the line crossed four edges, something is wrong.