Last reviewed 16 Sept 2026 · 7 min read
Classification of solids
Polyhedra (plane faces)
| Solid | Description |
|---|---|
| Regular polyhedra | All faces identical regular polygons — tetrahedron (4 equilateral triangles), cube/hexahedron (6 squares), octahedron (8), dodecahedron (12 pentagons), icosahedron (20) |
| Prism | Two equal and parallel polygonal bases joined by rectangular lateral faces; named by base shape (triangular, square, pentagonal, hexagonal prism) |
| Pyramid | Polygonal base and triangular lateral faces meeting at the apex |
Solids of revolution
| Solid | Generation |
|---|---|
| Cylinder | Rectangle revolving about one side |
| Cone | Right triangle revolving about one of its perpendicular sides |
| Sphere | Semicircle revolving about its diameter |
Other terms
- Right solid — axis perpendicular to the base; oblique solid — axis inclined to the base.
- Frustum — portion of a pyramid/cone between the base and a cutting plane parallel to the base.
- Truncated solid — cut by a plane inclined to the base.
- Generators — straight lines on the surface of a cylinder or cone (from base to apex for a cone).
- Slant height of a cone .
Positions of solids
| Position | Approach |
|---|---|
| Axis perpendicular to HP | Top view shows the true shape of the base; draw it first, project the front view |
| Axis perpendicular to VP | Front view shows the true shape of the base; draw it first, project the top view |
| Axis parallel to both HP and VP | Side view shows true shape of base; draw side view first |
| Axis inclined to HP, parallel to VP | Two-stage method — start with axis perpendicular to HP, then tilt the front view |
| Axis inclined to VP, parallel to HP | Two-stage — start with axis perpendicular to VP, then rotate the top view |
| Axis inclined to both HP and VP | Three-stage method — tilt, then rotate |
Resting conditions
- Resting on its base (on HP).
- Resting on a base edge or corner of the base with axis inclined.
- Resting on a lateral face or lateral edge (prisms/pyramids) or a generator (cones/cylinders).
- Suspended freely from a corner or point — the axis and the line joining the point of suspension to the centre of gravity are vertical (the CG of a pyramid/cone lies on the axis at 1/4 of the height from the base; of a prism/cylinder at mid-height).
Change of position method
Stage 1 (simple position): Draw the solid with its axis perpendicular to the plane to which it will be inclined — e.g. for an axis inclined to HP, draw with axis perpendicular to HP, placing the base edge/corner as required (e.g. edge perpendicular to VP).
Stage 2 (tilt): Redraw the front view tilted so that the axis (or base, face, edge) makes the given angle with HP; project the new top view using widths from stage 1 and positions from the tilted front view.
Stage 3 (rotate, if inclined to both planes): Redraw the top view rotated so that the axis (or its top view) makes the required angle with XY (φ or apparent angle β); project the final front view using heights from stage 2.
Key idea: during tilting about a line on HP, heights change but widths (from VP) of points measured perpendicular to the plane of rotation remain; during rotation about a vertical axis, heights remain unchanged.
Visibility
- In the front view, points nearest to the observer (farthest in front of VP) are visible.
- In the top view, points highest (farthest above HP) are visible.
- Edges hidden behind surfaces are drawn dashed; the outline of any view is always visible.
- For pyramids with apex up, all lateral edges are visible in the top view.
Useful geometric relations
- Regular tetrahedron of edge : height
- Square pyramid (base side , height ): slant edge ; slant height of a face
- Hexagonal base of side : across corners ; across flats
- Cone: slant height ; semi-apex angle
- Angle of a pyramid face with the base (square pyramid)
- Cube of side : face diagonal ; body diagonal