Chord Calculator: Circle Circumference

This calculator works out the circumference of a circle from its chord length and the chord’s distance from the center (or from the circle’s edge). Enter the chord length and the sagitta (the perpendicular distance from the chord midpoint to the arc), and you get the circle’s radius and circumference. It is useful for geometry problems, woodworking, metalworking, and any situation where you can measure a chord across a circular object but not its full diameter.

Formula

Given a chord of length c and a sagitta s (the perpendicular distance from the chord’s midpoint to the arc), the radius r of the circle is:

r = (s² + (c/2)²) / (2s)

Then the circumference C is:

C = 2πr

Where:

• c = chord length (same unit as sagitta)
• s = sagitta (distance from chord midpoint to the arc, perpendicular to the chord)
• r = radius of the circle
• π ≈ 3.14159

This formula comes from the intersecting chords theorem: if two chords intersect, the product of the segments of one equals the product of the segments of the other. Here, the chord is split into two equal halves, and the diameter is split into the sagitta and the remaining part (2r – s). So (c/2)·(c/2) = s·(2r – s), which rearranges to the formula above.

Worked Example

Using the default values from the calculator: chord length c = 10 cm, sagitta s = 2 cm.

1. Half the chord: c/2 = 5 cm.
2. Square it: (c/2)² = 25 cm².
3. Square the sagitta: s² = 4 cm².
4. Add them: 25 + 4 = 29 cm².
5. Multiply sagitta by 2: 2s = 4 cm.
6. Divide: r = 29 / 4 = 7.25 cm.
7. Circumference: C = 2 × π × 7.25 ≈ 45.55 cm.

The calculator shows radius 7.25 cm and circumference 45.55 cm, matching the manual calculation.

When to Use This Calculator

This method is ideal when you cannot measure the full diameter of a circular object because the center is inaccessible or the object is partially hidden. Common situations include:

• Measuring the circumference of a tree trunk or column by wrapping a tape around part of it.
• Determining the size of a circular arch or dome from a chord and rise.
• Working out the diameter of a pipe or wheel when only a segment is visible.
• Solving geometry homework problems that give a chord and its distance from the center.

If you can measure the chord and the sagitta accurately, you get the full circle’s dimensions without needing the center.

How to Read the Results

The radius is the distance from the center of the circle to any point on its edge. The circumference is the total distance around the circle. Both are in the same unit you used for the chord and sagitta (cm, inches, feet, etc.).

For example, if you measured in inches, the radius and circumference are in inches. If you measured in meters, they are in meters. The calculator does not convert units; it just applies the formula to whatever numbers you enter.

Note that the sagitta must be positive and less than the diameter. If the sagitta equals the radius, the chord length is zero (a point). If the sagitta is larger than the radius, the chord is imaginary – that combination is not possible for a real circle.

Common Mistakes and Misconceptions

One common mistake is mixing up the sagitta with the distance from the chord to the center. The sagitta is the distance from the chord midpoint to the arc (the curved edge), not to the center. If you have the distance from the center to the chord (call it d), then the sagitta is s = r – d, but you need the radius to use that. This calculator expects the sagitta directly.

Another mistake is using the chord length as the diameter. The chord is a straight line between two points on the circle; the diameter is the longest chord, passing through the center. Unless your chord goes through the center, it is shorter than the diameter.

Rounding too early can also cause errors. Keep several decimal places during intermediate steps, then round the final answer. The calculator rounds to two decimal places for display, but uses full precision internally.

Also, ensure the sagitta is measured perpendicular to the chord. If you measure at an angle, the value is wrong and the formula will not work.

Limits of the Formula

This formula assumes a perfect circle. Real objects may be slightly oval or irregular, so the result is an approximation for such shapes. For a very small sagitta relative to the chord, the radius becomes very large, and small measurement errors in the sagitta cause large errors in the radius and circumference. For example, if the sagitta is 0.1 cm and the chord is 10 cm, the radius is about 125 cm, but a 0.05 cm error in the sagitta changes the radius by about 62 cm. So this method is most accurate when the sagitta is not too tiny compared to the chord.

Also, the formula fails if the sagitta is zero (the chord would be a diameter, but then you already know the radius). In that case, the circumference is simply π × chord length.

Alternatives and Related Calculations

If you know the radius directly, the circumference is simply 2πr. If you know the diameter, it is πd. If you know the area, the circumference is 2π√(A/π).

For a chord and the distance from the center to the chord (not the sagitta), you can use the right triangle formed by the radius, half the chord, and the distance to the center: r² = (c/2)² + d². Then the sagitta is r – d. This calculator uses the sagitta directly, which is often easier to measure in practice.

If you have a circular segment (a slice of the circle), you might also want the arc length or the area of the segment. Those require the central angle, which can be found from the chord and radius using trigonometry.

Frequently Asked Questions

What is a chord in a circle?
A chord is a straight line segment whose endpoints lie on the circle. The diameter is the longest chord, passing through the center.

What is a sagitta?
The sagitta is the perpendicular distance from the midpoint of the chord to the arc (the curved edge) of the circle. It is sometimes called the “rise� of an arch.

Can I use this if I know the distance from the center to the chord?
Yes, but you need to convert. If you know the distance d from the center to the chord, then the sagitta is s = r – d. Since you don’t know r yet, you can use the formula r² = (c/2)² + d² to find the radius first, then the circumference.

Why does the formula give a huge radius for a tiny sagitta?
Because a nearly straight chord (small sagitta) implies a very large circle. For example, a chord of 10 cm with a sagitta of 0.1 cm gives a radius of about 125 cm. This is mathematically correct, but small measurement errors in the sagitta lead to large errors in the radius.

What units should I use?
Any consistent unit – centimeters, inches, feet, meters. Just make sure the chord and sagitta are in the same unit. The result will be in that same unit.

Is this formula exact for any circle?
Yes, for a perfect circle, the formula is exact. It comes from the intersecting chords theorem, which is a proven geometric relationship. The only approximation is in the value of π, which is irrational.

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in c | Chord length | cm | 10 | 0
in s | Sagitta (distance from chord midpoint to arc) | cm | 2 | 0
out r | Radius | cm | (s^2 + (c/2)^2) / (2*s) | 2
out C | Circumference | cm | 2 * pi * ((s^2 + (c/2)^2) / (2*s)) | 2