Article — Arch Calculator
Arch Calculator: Geometry of Span, Rise, and Curve
An arch's radius equals (S² + 4H²) / (8H), where S is span and H is rise. A 6 m span with 2 m rise produces a 3.25 m radius and a 7.60 m arc length, enough to frame a wide front door or a small bridge.
This calculator handles two of the three classical arch geometries: circular (constant radius) and parabolic (constant bending under uniform load). For each it returns radius, arc length, central angle, enclosed area, and rise-to-span ratio. Use it for masonry estimates, formwork layout, doorway design, or just to satisfy curiosity.
What is an arch?
An arch is a curved structural element that spans a gap by routing loads along its curve to the supports. Unlike a beam, which bends, an arch works primarily through compression. The masonry voussoirs (wedge-shaped stones) push against one another and against the abutments. As long as the line of thrust stays inside the arch, the structure holds.
Three numbers define an arch geometrically: span S (the horizontal distance between supports), rise H (the vertical height from the chord to the crown), and the curve type. Everything else — radius, arc length, area, thrust — derives from those three.
The Romans built more than 1,000 stone arch bridges across their empire. Some, like the Pont du Gard aqueduct, have stood without mortar for 2,000 years. The longest single span among them was the Trajan's Bridge on the Danube — 50 m of clear span, in the year 105 CE.
Arch types and their math
Six arch types appear in construction. They differ in how they distribute load and in how they look.
- Semicircular: H/S = 0.5. Easy to lay out, moderate thrust. Roman aqueducts.
- Segmental: H/S between 0.1 and 0.4. Less material, higher thrust.
- Pointed Gothic: Two arcs meeting at apex. Low thrust, tall structures.
- Parabolic: y = (4H/S²)·x·(S−x). Pure compression under uniform load.
- Catenary: Shape of a hanging chain inverted. Optimal under self-weight.
- Elliptical: (x/a)² + (y/b)² = 1. Used where headroom matters more than structural efficiency.
Arch calculator formulas
For a circular arch, the radius comes from a single line of algebra. The chord (span) and the sagitta (rise) together fix the radius of the underlying circle.
R = (S² + 4H²) / (8H) radiusθ = 2·arcsin(S / (2R)) central angleL = R · θ arc lengthA_seg = R²/2 · (θ − sin θ) segment areaFor a parabolic arch, no radius exists — the curvature changes along the length. Instead, arc length comes from a series expansion that converges quickly for shallow arches:
L ≈ S·(1 + 8/3·(H/S)²) arc length (1% error to H/S=0.5)A = (2/3)·S·H enclosed area (exact)slope = 4H/S tangent at supportsCircular vs. parabolic arch
The choice between circular and parabolic depends on aesthetics, load type, and ease of construction.
Circular arches dominate historical architecture. A compass and a string draw one. The radius stays constant, so identical voussoirs can be cast or cut in bulk. Roman, Romanesque, and Byzantine builders chose circular arches for nearly every structure.
Parabolic arches are the modern engineer's choice for uniform loads. Under a load that is constant per horizontal meter (snow on a barrel roof, water pressure on a dam face), the parabola routes everything through pure compression. Bending moments vanish. A circular arch under the same load develops bending in the haunches that requires extra cross-section to absorb.
Don't compute arch deflection with PL³/48EI. Arches deflect differently — through small support movements and curve flattening rather than mid-span sag. Use arch-specific analysis or finite element methods for accurate predictions.
Arch thrust and supports
Every arch exerts a horizontal thrust on its supports. The flatter the arch, the larger the thrust. For a circular or parabolic arch under uniform load w (force per unit horizontal length):
H_thrust = w · S² / (8H)
Halving the rise doubles the thrust. This is why Gothic cathedrals use buttresses — they convert the outward horizontal thrust of high vaults into vertical force at the foundation. Modern tied arches use a steel tension rod between supports to absorb thrust internally and free the foundation from horizontal loading.
If you can't size the abutments to absorb thrust, switch from a segmental to a pointed or stilted arch. Pointed arches at H/S = 1.0 produce roughly 25% less horizontal thrust than semicircular arches with the same span.
Worked arch examples
A garden gateway has a 1.2 m clear span and a 0.4 m rise. Radius = (1.44 + 0.64) / 3.2 = 0.65 m. Central angle = 2·arcsin(0.6 / 0.65) = 2.353 rad = 134.8°. Arc length = 0.65 · 2.353 = 1.53 m. The mason cuts 5 voussoirs of equal width and lays them on a wooden centering form.
A masonry bridge spans 8 m of river with a 1.6 m rise (segmental, H/S = 0.20). Radius = (64 + 10.24) / 12.8 = 5.80 m. Central angle = 2·arcsin(4 / 5.80) = 1.55 rad = 88.8°. Arc length = 9.0 m. Under a 30 kN/m roadway load, horizontal thrust = 30 · 64 / (8 · 1.6) = 150 kN at each abutment. The abutment must absorb that force or pull-back through tie-rods.
Common arch design mistakes
Five issues recur in arch projects, from garden gates to highway bridges.
- Ignoring horizontal thrust: Without proper abutments or tie-rods, supports move and the arch collapses.
- Confusing chord and arc length: Chord (span) is the straight line. Arc length is the curve. Order materials by arc length.
- Using beam deflection formulas: Arches deflect via support movement and curve change, not the beam equation.
- Insufficient self-weight check: The arch's own mass creates compression — usually a benefit, but skip the check and you may oversize.
- Asymmetric loading neglect: Snow on one side of a barrel roof creates bending the symmetric formulas miss. Use three-hinged analysis for safety.
- Wrong rise-to-span ratio for purpose: Too flat = thrust trouble. Too pointed = wasted material and headroom.
Historical arch ratios
The shape of an arch reveals the engineering knowledge of its builders.
The Gateway Arch in St. Louis (192 m span, 192 m rise, H/S = 1.00) follows a weighted catenary — designed so that under its own variable cross-section the entire structure carries pure compression. Eero Saarinen's team derived the equation in 1947 and built the arch in 1965. It remains the tallest free-standing arch in the world.