Arch Calculator — Circular Arch Geometry
Enter the horizontal span and vertical rise of a circular arch to find its radius of curvature, arc length, included angle, and enclosed area — useful for carpentry, masonry, bridge design and architectural drawing.
m
m
Length of the curved arch from one end to the other
- 1
Half-span
4 ÷ 2 = 2 - 2
Radius of curvature
(2² + 1,5²) ÷ (2 × 1,5) = 2,083 - 3
Half-angle α (rad)
arcsin(2 ÷ 2,083) = 1,287Half the angle subtended at the circle centre between the two spring points. - 4
Arc length
2 × 2,083 × 1,287 = 5,363
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For a circular arch with span S and rise h, the radius is R = (c² + h²)/(2h) where c = S/2. The arc length is 2R × arcsin(c/R) and the enclosed area is R²/2 × (θ − sin θ) where θ = 2 arcsin(c/R). Enter span and rise to get all properties instantly.
Képlet
How this is calculated
A circular arch is defined by its span (the horizontal chord between the two spring points) and its rise (the vertical height from the spring line to the crown). Given these two measurements, the radius of the underlying circle is uniquely determined: R = (c² + h²) / (2h), where c is the half-span. This formula comes from the intersecting-chords theorem of circle geometry.
Once the radius is known, the half-angle α = arcsin(c / R) gives the angular half-width of the arch. The full included angle is 2α. The arc length is simply the arc of a circle subtended by angle 2α: L = 2Rα. For a semicircular arch (rise = radius = span/2) this reduces to the familiar πR (half-circumference). The enclosed area under the arch — the circular segment — is A = R²/2 × (2α − sin 2α).
The rise-to-span ratio classifies the arch type: a ratio of 0.5 (rise = half the span) gives a semicircle; below 0.5 the arch is segmental (flatter than a semicircle); above 0.5 the arch is taller, approaching a pointed or horseshoe shape. The model assumes a perfectly circular arc — elliptical, parabolic or gothic arches use different formulas. All inputs are in the same unit (metres by default); outputs scale proportionally with any consistent unit.
Gyakran ismételt kérdések
Use R = (c² + h²) / (2h) where c is half the span and h is the rise. For example, a 4 m span (c = 2 m) with 1.5 m rise gives R = (4 + 2.25) / 3 = 2.083 m.
A segmental arch is a circular arch whose rise is less than the radius (i.e. rise-to-span ratio < 0.5). It is flatter than a semicircle and widely used in doorways, bridges and windows.
Once you have the radius R and half-angle α = arcsin(half-span / R), the arc length is 2Rα in radians. This calculator does all of that automatically from the span and rise.
Más néven
TG we-Calculate Editorial Team. (2026). Arch Calculator — Circular Arch Geometry [Online calculator]. TG we-Calculate. https://we-calculate.com/hu/calculator/arch-calculator
TG we-Calculate Editorial Team. "Arch Calculator — Circular Arch Geometry." TG we-Calculate. 2026. https://we-calculate.com/hu/calculator/arch-calculator.
TG we-Calculate Editorial Team, "Arch Calculator — Circular Arch Geometry," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/hu/calculator/arch-calculator
@misc{wecalculate_arch_calculator, title = {Arch Calculator — Circular Arch Geometry}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/hu/calculator/arch-calculator}}, year = {2026}, note = {TG we-Calculate} }
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