Calculate a Radius From a Straightedge and One Gap Measurement

The Problem

To rebuild a curved retaining wall with the same radius, you need that radius before you can order block — but the center point of the curve is somewhere out in the yard, unmarked and unreachable, and an overgrown wall may not let you measure across the top of the arc at all. All you can actually reach is the face of the curve itself.

The Technique

The chord-and-sagitta formula turns two measurements you can take from one side of any curve into its radius:

1. Hold a straightedge — a level works — against the curve so both ends touch it. The straightedge is your chord. Keep it positioned so both ends bear evenly.
2. Measure the gap between the straightedge and the curve at the widest point, which is the middle. That gap is the sagitta, or height of the arc.
3. Calculate: radius = (chord² ÷ (8 × gap)) + (gap ÷ 2).

Worked example from the source video, on a curved block wall: chord 12 inches, gap 3-1/4 inches. Chord squared is 144. Eight times the gap is 26. 144 ÷ 26 = 5.538, plus half the gap (1.625) gives a radius of 7.16 inches — a 14-5/16 inch diameter, which matched a direct tape measurement across the curve exactly.

4. On a real wall, take the pair of measurements at a few spots along the curve and average the results — built curves are never perfect arcs.