Developing Sectioned Pyramids and Cones

Developing Sectioned Pyramids and Cones

A truncated or sectioned pyramid or cone develops as the full solid's development with the cut marked on it — the top is then removed.

The procedure:

1. Develop the complete solid first: the fan of triangles for a pyramid, the sector for a cone.
2. On each generator of that development, mark the distance from the apex to the cutting plane, measured as a true length.
3. Join those marks with a smooth curve. The area between the base and that curve is the pattern for the truncated solid.

Getting the true distances. On a cone, only the outline generators show true length in the elevation. Every other generator's distance to the cut must be rotated onto an outline generator before it can be measured. Skipping that rotation is the standard error, and it produces a pattern that is systematically short.

Worked example. A cone cut at 45°: on the development the cut curve is not an arc. It is a curve whose distance from the apex varies smoothly, and it can only be plotted from rotated true lengths.

Core checkpoint: Develop the whole solid first, then mark the cut on it using rotated true lengths. Never try to develop the truncated shape directly.