The Trades Geometry Lab

You give it the two numbers you can always measure — the chord width of the opening and the rise you want at the peak — and it hands back everything the tape cannot: the radius of the circle that produces that exact arch, the center offset telling you precisely how far below the chord line to drive your trammel pin, the arc length so you know how much flexible trim or bender board to buy before you leave the supply house, and the central angle in degrees. The live blueprint redraws as you slide, so you can watch a shallow eyebrow arch flatten into a long radius or a tight half-round pull the pin right up to the chord.

It is the chord-and-sagitta relationship from circle geometry, rearranged for the field. In the formula, c is your chord (the full width of the opening) and r is the rise — the height of the arch at its center point. Half the chord and the rise form two known distances on a circle, and the intersecting-chords theorem locks in the only radius that can pass through both endpoints and the peak. Try a 36-inch chord with an 8-inch rise: the formula returns a 24.5-inch radius, which is a one-swing template for a classic arched doorway.

Both describe a perfect right angle — the difference is what a small measuring error does to each. If your mark is off by one sixteenth of an inch on a 5-foot hypotenuse, that error swings the corner by a measurable fraction of a degree; stretch the same triangle to a 15-foot hypotenuse and that identical sixteenth is spread across three times the distance, cutting the angular error to a third. Over a 40-foot foundation wall, that difference decides whether your far corner lands on the pin or an inch off it. The optimizer’s Accuracy Gain box computes this ratio live for your exact walls.

It tests both orientations of the triangle against your two wall lengths — 3k along the short wall with 4k along the long, and the reverse — finds the maximum multiplier k that fits each way, and recommends the larger result. It then converts the legs and hypotenuse into feet-and-inches you can pull directly on the tape, flags which standard trade sizes (3-4-5 through 18-24-30) fit inside your room, and gives you the rectangle diagonal √(L²+W²) as your final cross-check. Enter a 14-foot-9 wall against a 10-foot wall and it does in one keystroke what takes several minutes of longhand.

Completely free, no login, no account, no advertising — Global Sovereign University is a 501(c)(3) nonprofit and every tool it publishes is free forever. The lab is a single self-contained page: it loads once, runs entirely in your browser, and every calculation happens on your own device, so it keeps working in a basement or on a slab where the signal is one bar. It is built mobile-first — the sliders, number inputs, and live blueprints all work by thumb on a phone screen.

This lab is the working companion to Trades Algebra by Dr. Gene A Constant — the arc tool practices the geometry from the book and the optimizer practices the squareness math. You can read all eight chapters free in the GSU Reading Room, tap to listen to any chapter, download the seven free Blueprint Collection formula posters, and ask GENO — who has the entire book memorized — to walk you through any formula in 83 languages. The Kindle edition is $3.99 on Amazon.