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Radius from Chord and Height
Guide · Updated October 28, 2026
When you only have part of a circle, such as a broken pulley, a segment of a bore, or an arc on a casting, you can find the radius from two measurements: the chord c (straight distance across two points on the arc) and the height h, also called the sagitta (distance from the middle of the chord to the arc). The formula is R = c² ÷ (8h) + h ÷ 2.
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Step by step
- Measure the chord: Pick two points on the arc and measure straight across with a caliper or rule.
- Measure the height: At the midpoint of the chord, measure square to the chord up to the arc.
- Calculate: R = c² ÷ (8h) + h ÷ 2.
- Repeat: Take a second chord to confirm the arc is truly circular.
- Verify: Check with a radius gage or by sweeping an indicator if possible.
Worked example: segment of a bore
A broken flange has a chord of 2.000 in and a height of 0.125 in. R = 2.000² ÷ (8 × 0.125) + 0.125 ÷ 2 = 4.000 ÷ 1.000 + 0.0625 = 4.0625 in, so the original diameter was about 8.125 in.
A second chord of 3.000 in with h = 0.250 in gives R = 9.000 ÷ 2.000 + 0.125 = 4.625 in, which disagrees. That tells you the arc is not a true circle or one measurement is off; re-measure before making a part.
Sensitivity
Small heights magnify measuring error. With c = 2.000 in and h = 0.125 in, an error of 0.002 in in h changes R by about 0.06 in. Use the longest chord you can, and measure height with a depth mic or height gage on a surface plate instead of a rule.
Rearranged, the height for a known radius is h = R − √(R² − (c ÷ 2)²). For R = 4.0625 in and c = 2.000 in: 4.0625 − √(16.504 − 1) = 4.0625 − 3.9375 = 0.125 in, confirming the first example.
Worked example: metric pulley rim
A broken cast pulley rim measures a chord of 100 mm and a height of 10 mm. R = 100² ÷ (8 × 10) + 10 ÷ 2 = 10000 ÷ 80 + 5 = 125 + 5 = 130 mm, so the original outside diameter was about 260 mm.
Check with a second, longer chord: for R = 130 mm and c = 160 mm, the expected height is 130 − √(130² − 80²) = 130 − √(16900 − 6400) = 130 − 102.47 ≈ 27.5 mm. If you measure close to 27.5 mm, the radius is confirmed.
Quick shop check
Common problems: measuring height at a point that is not the chord midpoint, measuring to a burr or a chamfer instead of the true arc, and trusting a single chord.
FAQ
What is the sagitta?
The height of an arc above the middle of its chord.
What is the formula?
R = c² ÷ (8h) + h ÷ 2.
How accurate is it?
It is exact for a true circle; accuracy depends on how well you measure h.
How do I set a height for measuring?
Use gage blocks on a surface plate; build any stack free at gageblockcalc.com.
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