Circle Height Calculator: Find Rise from Width

This calculator works out the height (rise) of a circular segment when you know the width (chord length) and the radius of the circle. Enter the radius and the chord length, and you get the segment height, plus the arc length and segment area as useful extras.

Formula

The segment height (often called the rise or sagitta) is the distance from the midpoint of the chord to the arc, measured perpendicular to the chord. It is found with:

h = r – √(r² – (c/2)²)

Where:

• h = segment height (rise) – the answer you want
• r = radius of the circle
• c = chord length (the straight-line width of the segment)

This comes from the right triangle formed by the radius, half the chord, and the line from the circle’s center to the chord’s midpoint. That line has length √(r² – (c/2)²), and the height is the radius minus that length.

Worked Example

Using the default values from the calculator: radius r = 5 m, chord c = 6 m.

1. Half the chord: c/2 = 3 m
2. Square it: 3² = 9
3. Square the radius: 5² = 25
4. Subtract: 25 – 9 = 16
5. Square root: √16 = 4
6. Subtract from radius: 5 – 4 = 1

So the segment height is 1.000 m. The arc length works out to 2 × 5 × asin(6/10) = 10 × asin(0.6) ≈ 6.435 m, and the segment area is about 4.086 m².

When to Use This Calculation

This is the classic “sagitta� formula, used in many practical fields:

• Woodworking and construction: finding the rise of an arched frame, door, or window when you know the span and radius.
• Roofing and metalwork: laying out curved panels or gutters.
• Optics and mirrors: calculating the depth of a curved surface.
• Road design: determining the crown height of a curved road section.
• Any time you have part of a circle and need to know how “tallâ€� that part is.

The result is always positive and less than the radius. If the chord is very small compared to the radius, the height is tiny; if the chord approaches the diameter, the height approaches the radius.

How to Read the Result

The height h tells you the maximum distance from the chord to the arc, measured at the midpoint. This is not the same as the arc length (the curved distance) or the area of the segment. If you are using the height to build something, make sure you measure perpendicular to the chord, not along the arc.

For a semicircle (chord = diameter), the height equals the radius. For a very shallow segment (chord much smaller than diameter), the height is approximately c²/(8r) – a useful quick estimate when you don’t need precision.

Common Mistakes and Pitfalls

• Unit mix-ups: All inputs must be in the same unit. If you enter radius in meters and chord in centimeters, the calculator will give a wrong result. Convert first.
• Chord vs. arc length: The chord is the straight line across the segment, not the curved arc. Using the arc length instead of the chord will produce a nonsense height.
• Chord larger than diameter: If you enter a chord length greater than 2r, the formula breaks (you get a negative inside the square root). The calculator enforces a minimum radius of 0, but you must ensure c ≤ 2r.
• Rounding too early: If you round intermediate steps, the final height can be off. Keep full precision until the end.

Limits of the Formula

This formula assumes a perfect circle and a straight chord. Real-world objects may have slight imperfections, and if you measure the chord on a flexible material, it may not be exactly straight. Also, the formula gives the height for a single segment; if you have an arc that spans more than 180°, you are dealing with a major segment, and the height would be r + (distance from center to chord), not this formula. For a full circle, the height is 2r, but that is not a segment.

When using the calculator, always double-check that your chord length is less than or equal to the diameter. If you only know the arc length and radius, you need a different calculation (involving the angle).

Frequently Asked Questions

What is the difference between chord and arc?

The chord is the straight line connecting the two endpoints of the segment. The arc is the curved part of the circle between those same endpoints. The height is measured from the chord to the arc, not along the arc.

Can I use this for a half-circle?

Yes. For a semicircle, the chord equals the diameter (2r). Plugging that in gives h = r – √(r² – r²) = r, so the height equals the radius, which is correct.

What if I only know the height and chord, not the radius?

You can rearrange the formula to solve for the radius: r = (h² + (c/2)²) / (2h). This calculator does not do that directly, but you can use the formula manually or look for a “radius from chord and height� calculator.

Why is my result different from measuring it physically?

Measurement error is common. Check that your chord is measured straight across, not along a curve, and that your radius is accurate. Also, if the object is not a perfect circle, the formula will not match.

Does the calculator work for any unit?

Yes, as long as you use the same unit for both inputs. The output will be in that same unit. For example, if you enter feet, the height will be in feet.

What is the maximum height possible?

The maximum height for a segment is the radius, which occurs when the chord is the diameter. If you enter a chord equal to 2r, you get h = r. If you enter a larger chord, the formula is invalid.

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in r | Radius | m | 5 | 0
in c | Chord length | m | 6 | 0
out h | Segment height | m | r - sqrt(r^2 - (c/2)^2) | 3
out arc | Arc length | m | 2 * r * asin(c / (2 * r)) | 3
out area | Segment area | m² | r^2 * acos(1 - h/r) - (r - h) * sqrt(2*r*h - h^2) | 3