Circular Segment Calculator (Arch from Span and Rise)
Radius, arc length, central angle and area of a circular arch or curved window from the two measurements you can actually take: the span and the rise.
Results
Radius of the arc
1.4500
m
Arc length
2.2069
m
Central angle
87.206
°
Segment area
0.5500
m²
Segment perimeter (arc + chord)
4.2069
m
What this tool does
On site you can measure an arch's span and how high it rises, but not its radius — the centre is somewhere in mid-air. Two measurements are enough: the radius is (c²/4 + s²) ÷ 2s, the old carpenter's rule for setting out an arch. The angle is computed as 4·atan(2s/c), which stays correct even for arches taller than a semicircle; the drawing shows the whole sector the segment belongs to.
Formula
R = (c² ÷ 4 + s²) ÷ (2s) · θ = 4 atan(2s ÷ c) · arc = R θ · A = R² (θ − sin θ) ÷ 2
Variables
| Symbol | Meaning | Unit |
|---|---|---|
c | Chord (straight span) | m |
s | Rise (height of the arc) | m |
R | Radius of the arc | m |
AL | Arc length | m |
CA | Central angle | ° |
AR | Segment area | m² |
PE | Segment perimeter (arc + chord) | m |
Worked example
- Chord (straight span)2 m
- Rise (height of the arc)0.4 m
- Radius of the arc1.4500 m
- Arc length2.2069 m
- Central angle87.206 °
- Segment area0.5500 m²
- Segment perimeter (arc + chord)4.2069 m
Limitations
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.