Arch Calculator

An arch's geometry is fully determined by just two measurements — span (the horizontal opening width) and rise (the vertical height from the springline to the arch's highest point) — but the relationship between those two numbers and the arch's actual radius is not linear, which is why a segmental arch calculator is genuinely useful rather than a simple lookup. The formula R = (S²/8H) + H/2 comes directly from the geometry of a circular segment: as rise shrinks relative to span, the implied radius grows rapidly, meaning a very shallow (low-rise) arch requires a surprisingly large radius circle to generate its curve — a detail that catches out anyone trying to lay out a shallow arch by eye or with a fixed-length string and stake.\n\nThe distinction between semicircular, segmental, and pointed arch types matters structurally as well as visually. A semicircular (Roman) arch has rise equal to half the span and directs load straight down through the haunches into the supporting piers. A segmental arch — shallower, with rise less than half the span — is common over doors and windows where full headroom below a full semicircle isn't needed, but it generates more outward horizontal thrust at the springline that the supporting wall or abutment must resist. A pointed (Gothic) arch redirects load more steeply downward, which is part of why it enabled taller, thinner-walled Gothic structures than round-arch Romanesque construction could achieve.

The true arch, built from wedge-shaped voussoirs that carry load in compression down to abutments, appears in Mesopotamian and Egyptian brickwork in the second millennium BCE, but Roman engineering made it standard practice. Roman builders applied the semicircular arch systematically across aqueducts, bridges, and amphitheatres, using timber centering to hold the voussoirs until the keystone was set; the Colosseum arcades, completed in 80 CE, are the familiar demonstration. The shallower segmental arch was in use earlier than European practice alone would suggest: the Anji Bridge in Hebei, China, built by the engineer Li Chun around 595 to 605 CE, spans roughly 37 metres and remains the oldest known open-spandrel segmental arch bridge. The pointed arch developed in Islamic architecture from the eighth century and reached Western Europe by the twelfth, where the choir rebuilt at the Abbey of Saint-Denis under Abbot Suger between 1140 and 1144 is conventionally treated as the beginning of Gothic construction. Analysis lagged practice by centuries. Robert Hooke stated the governing principle in 1675 as a Latin anagram, later rendered as the observation that as hangs the flexible chain, so but inverted stands the rigid arch - the catenary as the ideal line of compression. Giovanni Poleni used a hanging-chain model in 1748 to assess the cracked dome of St Peter's in Rome, and Jacques Heyman's work from the 1960s onward formalized masonry arch safety, showing that an arch stands if any line of thrust fits within its thickness.