Catenary Arch Calculator

Results

Derived parameter a

1.263497 m

Arc length S

3.756059 m

Area under arch

0.955278 m²

Arch ring volume (approx)

0 m³

Estimated ring weight

0 kg

#	x (m)	y (m)
0	-1.5	1
1	-1.38	0.831583
2	-1.26	0.682078
3	-1.14	0.550136
4	-1.02	0.434566
5	-0.9	0.334323
6	-0.78	0.248504
7	-0.66	0.176334
8	-0.54	0.117161
9	-0.42	0.070451
10	-0.3	0.035783
11	-0.18	0.012843
12	-0.06	0.001425
13	0.06	0.001425
14	0.18	0.012843
15	0.3	0.035783
16	0.42	0.070451
17	0.54	0.117161
18	0.66	0.176334
19	0.78	0.248504
20	0.9	0.334323
21	1.02	0.434566
22	1.14	0.550136
23	1.26	0.682078
24	1.38	0.831583
25	1.5	1

Span	Rise	Points	Thickness	Depth	Density
2 m	0.6 m	21	0.25 m	0.6 m	2400 kg/m³
3.5 m	1 m	41	0.3 m	0.8 m	2400 kg/m³
8 m	2 m	61	0.45 m	1 m	2400 kg/m³
26.247 ft	6.562 ft	61	1.476 ft	3.281 ft	150 lb/ft³

Formula used

The catenary is defined by y(x) = a cosh(x/a) - a, centered at the crown. For a symmetric opening of span W and rise H, parameter a satisfies:

a(cosh(W/(2a)) - 1) = H

Arc length between springings: S = 2a sinh(W/(2a))
Area under the curve: A = 2a² sinh(W/(2a)) - aW
Coordinates from crown to springing: (x, y) = (x, a cosh(x/a) - a) for x ∈ [-W/2, W/2]

We solve the equation for a with robust bisection. Volumetric estimate for an arch ring uses V ≈ S · t · d, where t is thickness and d is out-of-plane depth.

How to use

Select your preferred units.
Enter span and rise of the opening.
Optionally set points, thickness, depth, and density.
Click Calculate to generate geometry and quantities.
Use Download CSV for coordinates and summaries.
Use Download PDF to archive a compact report.

This tool provides geometric quantities and simple estimates. It is not a substitute for professional structural design or checks.

FAQs

What shape does this tool compute?

It computes a true catenary: y = a cosh(x/a) − a. This differs from a parabola, especially at larger spans. The solver fits “a” from span and rise, then derives length, area, and coordinates.

When should I use a catenary arch?

Use a catenary when the arch primarily carries uniform self‑weight along its curve, such as masonry or compression-only shells. It approximates thrust lines well, reducing bending. Always verify with structural design checks.

How accurate are the results?

We solve for a with bisection to a tight tolerance. Geometry is exact for the catenary model; volumes and weight use simple ring assumptions. Input rounding and units choice influence displayed precision.

What input ranges are recommended?

Span and rise must be positive. Rise typically remains smaller than span/2 for practical openings. Very extreme ratios may be unrealistic. Points controls smoothness only; higher values produce denser coordinates without changing geometry.

What units and densities are supported?

Choose metric or US customary for length and density. Internally we calculate in SI, then convert. Default density suits concrete; replace it for brick, stone, or steel to estimate ring weight.

How can I export or share results?

Use Download CSV for point coordinates and summaries. Use Download PDF for a compact report. The Shareable Link button reloads the same inputs, helping you document design assumptions or send setups to teammates.

Is this a replacement for structural design?

No. This calculator provides geometric quantities and rough weights only. It does not check stresses, stability, or support conditions. Engage a qualified engineer to design reinforcement, bearings, and safety factors.