Sections of Right Regular Solids

Main Idea

A section is the shape revealed when a solid is cut by an imaginary plane. For a right regular solid, a vertical cutting plane is perpendicular to the horizontal plane and may be parallel or inclined to the vertical plane. The resulting sectional shape depends on the type of solid, the position of the cutting plane, and the edges, faces, or generators that it intersects. Orthographic projections are used to represent the true section, the remaining portion of the solid, and the cut surfaces accurately.

Key Concepts and Definitions

• Right Regular Solid: A solid having a regular plane figure as its base and an axis perpendicular to the base, such as a prism or pyramid.
• Section Plane: An imaginary plane used to cut a solid so that the shape of the cut surface can be studied.
• Vertical Cutting Plane: A section plane perpendicular to the horizontal plane. It may be parallel or inclined to the vertical plane.
• Sectional Elevation: The front-view projection of a solid after it has been cut by the section plane.
• Sectional Plan: The top-view projection showing the position of the cutting plane and the points where it intersects the solid.
• True Shape of Section: The actual shape of the cut surface, obtained on an auxiliary plane parallel to the section plane.
• Cutting Trace: The line in a projection that represents the intersection of the cutting plane with the corresponding reference plane.
• Section Points: Points where the cutting plane meets the edges, generators, or faces of the solid.
• Sectional Surface: The new plane surface exposed by cutting the solid.
• Hatching: Thin, evenly spaced lines drawn on the cut surface, commonly at about 45 degrees, to identify the sectional area.
• Prism: A solid with two equal and parallel polygonal bases connected by rectangular or parallelogram faces.
• Pyramid: A solid with a polygonal base and triangular faces that meet at one vertex called the apex.
• Cylinder: A solid with two equal, parallel circular bases connected by a curved surface.
• Cone: A solid with a circular base and a curved surface that meets at one point called the vertex.

Supporting Arguments and Evidence

• A vertical section plane is perpendicular to the horizontal plane. Consequently, its trace on the horizontal plane is generally used to locate section points in the plan. If the cutting plane is parallel to the vertical plane, its vertical trace is represented by a line parallel to the XY reference line in the elevation.

• The section depends on the exact intersection of the cutting plane with the solid. For a prism, the section is formed by joining the points where the plane cuts the lateral edges or faces. The resulting polygon may have fewer sides than the base, depending on the position of the plane.

• In a pyramid, section points are located on the slant edges or triangular faces. These points must be joined in their correct sequence to form the sectional polygon.

• For a cylinder, a vertical plane parallel to the axis produces a rectangular section. A plane passing through the axis produces a rectangle whose width equals the diameter.

• For a cone, a vertical plane through the axis produces a triangular section. A vertical plane not passing through the axis generally produces a portion of an ellipse or another curved section in the true shape.

• Accurate representation requires the section points to be joined in their actual order around the cut surface. Points in the plan and elevation correspond through projectors, and their positions are controlled by the section plane.

• The apparent shape in the front or top view is not necessarily the true shape of the section unless the section plane is parallel to that projection plane. The true shape is obtained by projecting the section points perpendicular to the section plane onto an auxiliary reference line or auxiliary plane parallel to the section plane.

• The usual construction sequence is to draw the projections of the solid, mark the cutting plane, locate the section points, project corresponding points, join them in order, show visibility, and hatch the sectional surface.

• A clear engineering drawing distinguishes the remaining solid, the sectional boundary, hidden details, and the cut surface. Visible portions are drawn with continuous thick lines, hidden portions with dashed lines where required, and the cut surface is identified using uniform hatching.

• Section lines should be thin, evenly spaced, and generally inclined at 45 degrees, unless this direction would make them parallel to an important edge. The shortest distance between a point and a plane is measured along a perpendicular to that plane.

• Accurate construction of the base is essential for correct sectioning. For a regular polygonal base, the base angles and side relationships are equal according to the type of polygon. The axis of a right regular solid is perpendicular to its base and passes through the centre of the regular base.

What to Remember

To section a right regular solid with a vertical cutting plane, project the solid accurately, locate every intersection point, join the points in their correct order, and distinguish visible, hidden, and sectional features using the correct line conventions. When the cutting plane is inclined to both principal reference planes, use an auxiliary projection parallel to the cutting plane to obtain the true shape of the section.