Arch Calculator

Calculate circular arch radius, arc length, segment area, and material volume from span, rise, and depth.

Calculate arch dimensions
Enter all dimensions in one consistent length unit.

About the arch calculator

A circular arch is defined conveniently by its span and rise. The span is the straight chord between the springing points, and the rise is the perpendicular distance from the chord midpoint to the crown. Those two field measurements determine the radius and central angle of the circular segment. This calculator then finds the curved arc length, the enclosed segment area, and the volume produced when that area extends through a specified depth. The radius formula is span squared divided by eight times the rise, plus half the rise. After finding the radius, the calculator derives the central angle from the chord relationship. Arc length equals radius multiplied by the central angle in radians. Segment area equals one half of radius squared multiplied by the central angle minus its sine. Finally, material volume equals segment area multiplied by arch depth. A semicircle with a 10-unit span and 5-unit rise therefore has a 5-unit radius, a 15.708-unit arc, a 39.27-square-unit area, and 78.54 cubic units at a depth of 2. These measurements are valuable for preliminary masonry, formwork, trim, window, doorway, tunnel, and decorative feature estimates. Arc length can help estimate curved trim or centering length. Segment area helps compare the opening or infill area, while volume gives a simple extrusion estimate for a solid segment. Real masonry arches often use rings, voussoirs, joints, or a hollow opening, so the gross segment volume is not automatically the final quantity of brick, stone, concrete, or mortar. Measure the clear span between springing points and measure rise at the centerline. Keep span, rise, and depth in the same unit. The circular-segment model supports semicircular and shallow segmental arches, but pointed, elliptical, parabolic, and multi-centered arches follow different geometry. Structural capacity also depends on loading, support, material strength, ring thickness, and workmanship. Use the results for geometry and estimating, then have a qualified designer verify load-bearing construction.

Arch calculation examples

Circular segment examples for common preliminary layouts.

DimensionsCalculated geometryUse
Span 10 ft, rise 5 ft, depth 2 ftRadius 5 ft; volume 78.54 ft3A full semicircular segment.
Span 8 ft, rise 2 ft, depth 1 ftRadius 5 ft; volume 11.182 ft3A shallow segmental arch.
Span 6 m, rise 3 m, depth 0.4 mArc 9.425 m; volume 5.655 m3A semicircular architectural feature.

How to use the arch calculator

  1. Measure the clear horizontal span between the two springing points.
  2. Measure the vertical rise from the span midpoint to the crown.
  3. Enter the front-to-back depth using the same unit.
  4. Select Calculate Arch and review radius, arc length, area, and volume.

Arch calculator FAQ

What arch shapes does this calculator support?

It models a circular segment, including a semicircular or shallow segmental arch. Pointed, elliptical, and parabolic arches require different formulas.

What is the difference between span and arc length?

Span is the straight horizontal chord between springing points. Arc length follows the curved face from one springing point to the other.

Does material volume include waste?

No, the volume is the exact geometric extrusion of the segment area. Add an allowance appropriate for cutting, joints, breakage, and site conditions.

Can the dimensions be entered in meters?

Yes, meters, feet, inches, or another unit can be used. Keep every input consistent so area and volume inherit the correct squared and cubed units.

Can I use this result for structural design?

The calculator provides geometry only and does not check thrust, loading, reinforcement, or support capacity. A qualified engineer should verify load-bearing arches.