Finding the Center of an Arc

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.

The Problem

Someone hands you a cardboard template, a curved piece of trim, or a pattern with an arc and says "I need this exact radius." You don't have the architectural plans or any measurements. You just have the curve and need to find the center point and radius so you can reproduce it.

The Technique

This is a geometry trick called the perpendicular bisector method. It works on any arc from any circle, every time.

1. Mark three points anywhere along the arc — call them A, B, and C. Space them out for best accuracy.
2. Draw a straight line from A to B, and another from B to C
3. Find the midpoint of line AB. Mark it.
4. Draw a line perpendicular (90 degrees) to AB from that midpoint, extending it well past the arc
5. Find the midpoint of line BC. Mark it.
6. Draw a line perpendicular to BC from that midpoint, extending it the same way
7. Where those two perpendicular lines cross — that's the center of the circle
8. Measure from that intersection to any of your three original points — that's the radius

Set your compass, dividers, or trammel to that radius, swing from the center point, and you'll trace the exact arc.