Graspline

Projections of solids: a solved example

A pentagonal pyramid of 30 mm base side and 60 mm axis rests on HP on one of its base edges with its axis inclined at 30° to HP and parallel to VP. Draw its projections.

Every line below was drawn and checked by Graspline's own geometry engine, by the textbook method, one step at a time with the reason for each.

XYabcdeo30a′(b′)e′(c′)d′o′60a′(b′)e′(c′)d′o′60°θabcdeo
The finished drawing.

The construction, step by step

  1. Draw the reference line XY

    Pentagonal pyramid of 30 mm base side and 60 mm axis, to rest on HP on its base edge ab with the axis at θ = 30° to HP (base at 60°). The axis ends up inclined to HP, so use the change-of-position method: first draw the solid with its axis perpendicular to HP, then tilt it.

  2. Stage 1: top view with the axis perpendicular to HP

    With the axis perpendicular to HP, the top view is the true shape of the base with the apex o at its centre. Orientation matters: the base edge ab is placed perpendicular to VP (a line perpendicular to XY here, a point in the front view), because that is the element the solid will tilt about. Hidden edges here: none.

  3. Stage 1: project the front view

    Project every corner. The base is on XY; the axis is 60 mm long. Edges on faces turned away from the viewer are hidden: none.

  4. Stage 2: redraw the front view tilted

    Reproduce the stage-1 front view turned about a′ (the base edge ab stays on HP) so that the axis makes θ = 30° with HP (base at 60°). Every length in this view is unchanged — copy them with dividers.

    Hidden edges in this view: none.

  5. Stage 2: project the new top view

    Tilting about a line perpendicular to VP leaves every point's distance from VP unchanged. Draw horizontal loci from the stage-1 top view points; the projectors from the tilted front view cut them at the new points. Join the corners in the same order as before.

    Visibility: a face is seen in this view if its outside faces the viewer. Hidden edges: ab, bc, ea, ao, bo (dashed).

  6. Result

    The final stage-2 views are the required projections. Dashed edges are hidden.

    Axis inclination to HP30°
    Axis inclination to VP0°
    Highest point above HP47.9 mm
    Length of axis in top view52 mm
    Length of axis in front view60 mm

    ✓ Verified: the base edge ab lies on HP, the axis inclination is as required, and all edge lengths are preserved in 3D (max deviation 0 mm).

Draw your own question in Graspline →

Free while we test it.

Graspline is not affiliated with any university or examination board. Terms of use