An arch pushes outward on its supports as well as down. For the same load and span the horizontal push grows as the rise falls: halve the rise and you roughly double the thrust.
H ≈ w·s² ⁄ 8f (uniform load w per metre, span s, rise f)
Rules · Proportion · Geometry
The √3 rectangle and the equilateral triangle
Swing the diagonal of a √2 rectangle to obtain a rectangle of 1 : √3, the proportion of an equilateral triangle’s height to half its base, used for setting out by triangles (ad triangulum).
√3 ≈ 1.732; height of an equilateral triangle of side 2 = √3
The series of root rectangles (√2, √3, √4 = 2, √5) can be generated one from another with a compass, each new diagonal swung onto the side. Of these the √3 rectangle is tied to the equilateral triangle and the hexagon, which gave its name to the “ad triangulum” method debated at Milan Cathedral in the 1390s, when the height of the church was argued between square-based and triangle-based schemes.
Related rules
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
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
Cite this page Chris Paton, “The √3 rectangle and the equilateral triangle”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/root-three-rectangle. Text, drawings and models licensed under CC BY-NC 4.0.