BIZOMAT MAKEOS · Geometry
Circular Segment and Arc
What chord, arc length, angle and area does this circular segment have?
Solve a circular segment from its radius and height, or find the radius of an arc from a measured chord and height.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Known values | choice | Choose what you have measured or know from the drawing. |
| Radius | mm | Drawing radius (used only with radius and height). |
| Chord length | mm | Straight distance between the arc ends, measured or from the drawing (used only with chord and height). |
| Segment height (sagitta) | mm | Distance from the chord midpoint to the arc, at right angles to the chord. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Known values
- What it is
- Which pair of values you already have. With radius and height the tool solves the segment; with chord and height it works out the radius of the arc first, which suits measuring an existing curved part.
- Units and format
- Choose one of 2 options from the list: Radius and segment height; Chord and segment height (find the radius).
- Example
- The form opens with “Radius and segment height” selected as a worked example. Choose the option that matches your case.
- Get it right
- Only the inputs for the chosen pair are used; the other value is ignored, so a stale number there does no harm.
What this means: Radius
- What it is
- The radius of the circle the arc belongs to, from the arc centre to the curved edge, as given on the drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the radius, not the diameter: halve a diameter first. This value is ignored when the chord and height pair is selected.
What this means: Chord length
- What it is
- The straight-line distance between the two ends of the arc, across the open side of the segment.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only with the chord and height pair. Measure end to end in a straight line, not along the curve.
What this means: Segment height (sagitta)
- What it is
- The height of the segment (sagitta): the distance from the middle of the chord to the arc, measured square to the chord.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure at the exact midpoint of the chord. When the height is small compared with the chord, a small error in it changes the calculated radius a lot.
Method and formulas
- From chord c and height h: radius r = (c²/4 + h²) ÷ (2h).
- Included angle θ = 2 × arccos((r − h) ÷ r); chord = 2√(h(2r − h)); arc = r × θ (θ in radians).
- Segment area = r²(θ − sin θ) ÷ 2.
- Example: radius 50 mm and height 20 mm give a chord of 80 mm, an included angle of 106.26°, an arc of 92.73 mm and an area of 1,118.2 mm².
Limitations
- Assumes a true circular arc; a measured profile with form error gives an averaged radius.
- A small height relative to the chord makes the calculated radius very sensitive to the height measurement.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the radius of a curved part from a chord and height measurement
- Lay out arcs, flats and partial bores
- Estimate area of a segment-shaped plate or liquid section
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- Radius
- Chord length
- Included angle
- Arc length
- Segment area
Interpreting the result
- Check the sensitivity table: when the height is small, a few micrometres change the radius considerably.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 77–80. Topic reference; no source tables or recommended settings are reproduced.
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 66–76. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Large arcs, ellipses and curves by points
What it covers: Find a radius from chord and rise, mark arcs whose centre is off the sheet by ordinates, find an arc's centre, and draw true ellipses with string, trammel and offsets.
Method explained here · Library collection
Circles, arcs, chords and bolt circles: hole patterns, rolled shells, bends and radius checks
In Practical Mathematics for Engineering and Manufacturing
What it covers: How circumference, arc length, chords and sagitta solve practical problems, from bolt-circle hole coordinates and rolled cylinders to sheet metal bend allowances and finding a radius from a chord.
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