A chain hanging freely takes the one shape in which it is in pure tension; turn that shape upside down and it is the shape of an arch in pure compression for the same loads.
free-hanging chain (catenary) ↔ ideal arch under its own weight
Name a roof by its slopes
Classify a roof by how its planes meet: one slope (shed), two meeting at a ridge (gable), four meeting at hips (hip, pyramid), a hip below a gable (hip-and-gable), two pitches each side (gambrel, mansard), or a curved surface (barrel, cone, dome).
Every roof form is an answer to three questions: how wide is the span, how much water or snow must it shed, and what should it say? The gable is the cheapest way to span a rectangle; the hip protects all four walls and needs no gable walls; the pyramid and cone crown centralised plans; the sawtooth admits north light to a factory floor; the butterfly collects rain in a central gutter. Cultures chose among the same few geometries and made them their own by pitch, curvature and covering.
Related rules
Draw a pointed arch from two centres on the springing line. With the centres at the springings, the arch is equilateral; with centres beyond them, a lancet; with centres between them, a drop arch. Late Gothic added arches of four centres and the reversed curve of the ogee.
r = s (equilateral) · r > s (lancet) · s⁄2 < r < s (drop)
Classify arches by the geometry of the intrados: the number and position of the centres (one for round and segmental, two for pointed, three or four for basket and Tudor arches), or a single continuous curve (ellipse, parabola, catenary), and distinguish true arches, with voussoirs, from corbelled “arches” of oversailing courses.
A round arch must rise half its span, so arches of different spans on one springing line have crowns at different heights. A pointed arch can be given any rise at least half its span, so arches of different widths can all be made to reach the same height.
round: f = s⁄2 (fixed) · pointed: r = (s²⁄4 + f²) ⁄ s for any f ≥ s⁄2
To find how thick the pier of an arch must be, divide the intrados into three equal parts; draw a line from the springing through the first division, and produce it beyond the springing by its own length. A vertical through the end of that line marks the outer face of the pier.
pier ≈ ¼ span (semicircle) · ≈ 0.23 span (equilateral) · less for a lancet
To combine the unity of a central plan with the direction a church needs, plan the space as an oval, set across the axis to widen the view of the altar, or along it to lead the eye towards it.
Cite this page Chris Paton, “Name a roof by its slopes”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/roof-forms-gallery. Text, drawings and models licensed under CC BY-NC 4.0.