Developing Pyramids and Cones

Developing Pyramids and Cones

Both develop as a fan of triangles about a single centre — the apex — because every surface line runs to the apex.

A right pyramid. Find the true length of a sloping corner edge by rotation. Swing an arc of that radius from a centre; step the true base side lengths around it as chords; join each division to the centre. The base sides are true in the plan, so they transfer directly.

A right cone. All generators are equal, so the development is a sector of a circle. Its radius is the true generator length — the slant height, taken from the outline of the elevation. The sector angle is

θ = 360° × r ÷ L

where r is the base radius and L the slant height.

Worked example. A cone of base radius 30 mm and slant height 90 mm develops to a sector of 360 × 30 ÷ 90 = 120°. Using the vertical height instead of the slant height is the classic error — it gives a sector that is too wide, and the rolled cone comes out the wrong angle.

Core checkpoint: Use the slant height as the sector radius, never the vertical height. They are different lengths and only the slant lies on the surface.