Library / Geometry, coordinates, surveying and trigonometry / GEO-07
Geometry, coordinates, surveying and trigonometry
Rise of a minor circular arc
Height from the midpoint of a chord to the minor arc above it, from the chord length and the radius.
- Ref
- GEO-07
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- h [cm]
The rise (or sagitta) is the greatest distance from a chord to its minor arc. It equals r − √(r² − c²/4). The calculator uses the equivalent form c²/[4(r + √(r² − c²/4))], which avoids subtracting two nearly equal numbers when the arc is shallow. The chord must not exceed the diameter 2r.
Height from the midpoint of a chord to the minor arc above it, from the chord length and the radius. This rationalised form avoids the loss of precision on shallow arcs.
Calculator
- Distance from the centre to the chord, r − h3 cm
- Central angle, 2 arcsin(c/2r)106.3 °
- Rise by the approximation c²/(8r)1.6 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- An 8 cm chord of a 5 cm circle has its arc rising 2 cm above it, with the centre 3 cm below the chord.
- For a shallow arc the quick estimate c²/(8r) = 1.6 cm is close; here it is 20 % off, so use the exact form for larger arcs.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| c Chord length (up to 2r) | 0.5 to 10 cm | 0.006254 | 5 | 4.994 | +10% | +31.3 % |
| r Radius | 4 to 30 cm | 0.2679 | 4 | 3.732 | +10% | −13.7 % |
Sweep of Chord length (up to 2r)
| Chord length (up to 2r) (cm) | Rise of the arc (cm) |
|---|---|
| 0.5 | 0.006254 |
| 1.688 | 0.07171 |
| 2.875 | 0.2111 |
| 4.063 | 0.4312 |
| 5.25 | 0.7445 |
| 6.438 | 1.174 |
| 7.625 | 1.765 |
| 8.813 | 2.637 |
| 10 | 5 |
Sweep one variable
Variables and how each one contributes
Rise of the arc
What the equation gives you. Everything else feeds it.
Radius
The radius of the circle, in cm. It must be at least half the chord. A larger radius makes the arc flatter, so the rise falls. Raising it by 10 % (to 5.5) moves the result by −13.7 %, from 2 to 1.725 cm.
Chord length (up to 2r)
The length of the chord, in cm, from 0 to the diameter 2r. A longer chord raises the rise, and the rise grows faster than the chord does. Raising it by 10 % (to 8.8) moves the result by +31.3 %, from 2 to 2.625 cm.
Worked example with the values above
- Half the chord squared: c²/4 = 16; r² = 25.
- Root: √(25 − 16) = 3, so the denominator is 4(5 + 3).
- h = 64 / 32 = 2 cm.
Assumptions and limits
- A true circle, with angles in radians where the formula uses them.
- Lengths are measured from the same centre, in one consistent unit.
Common mistakes
- Entering a central angle in degrees where radians are needed.
- Using the full chord in place of the half chord when working with the right triangle.
- Using the major arc when the minor arc is meant.
Statistics and uncertainty
Circle results depend on the radius squared or on trigonometric functions of the angle, so errors in the radius are doubled in areas. The segment and rise formulas can lose digits for very small angles or very shallow arcs.
Where it is used
- Cutting, rolling and setting out curved plates, pipes and arches.
- Partly filled pipes and tanks, washers and flanges, and tangent lines past a circular object.
Maths you need
- Radians: angle measured as arc length over radius
- Right triangle: Pythagoras and sine over hypotenuse
- Circle area: π times radius squared