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 diagonal of the square
To make a rectangle of 1 : √2, draw a square and swing its diagonal down to extend one side; the result cannot be measured in whole units of the square, but it can be drawn exactly.
√2 ≈ 1.414; halving a 1 : √2 rectangle gives two more of the same shape
Palladio includes the √2 room among his seven (The seven most beautiful room shapes); medieval masons used the same construction, rotating squares within squares (ad quadratum), to set out pinnacles and elevations. The rectangle has a unique property: cut in half across its long side it makes two rectangles of the same proportion. That is why it was chosen in 1922 for the German paper standard that became the international A-series; an A4 sheet is 210 × 297 mm.
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 diagonal of the square”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/sqrt2-rectangle. Text, drawings and models licensed under CC BY-NC 4.0.