Arch Calculator

Calculate arch radius, curve lengths, opening area, ring area, and masonry volume from key arch dimensions. Arch Calculator helps builders, masons, engineers, and homeowners plan and estimate curved structural openings.

Highlights

  • Calculate accurate arch radii from span and rise dimensions.
  • Estimate inner and outer curved arch lengths for construction planning.
  • Determine opening area and arch ring face area.
  • Estimate masonry volume using wall depth and ring thickness.
  • Supports metric and imperial units for flexible project planning.

Table of Contents

  1. How Arch Calculator Works?
  2. Practical Use Cases
  3. Accuracy and Limitations
  4. Quick Examples
  5. Frequently Asked Questions (FAQs)
  6. Cite This Calculator

How Arch Calculator Works?

The Arch Calculator models the clear opening as a circular segment. It uses the opening span, arch rise, arch ring thickness, and wall depth to estimate the geometry of the inner arch, the surrounding arch ring, and the required masonry volume.

Let s represent the opening span and h represent the arch rise. The intrados radius is calculated from the geometry of a circular segment:

R=s28h+h2

Here, R is the intrados radius. The included central angle of the arch is determined from the half-span and radius:

θ=2sin−1(s2R)

The calculator uses this angle to find the curved lengths of the inner and outer arch surfaces. The intrados length is Rθ. The extrados radius is obtained by adding the radial arch ring thickness t:

Ro=R+t
Li=Rθ,Lo=Roθ

The clear opening area is the area of the circular segment beneath the intrados curve:

Ao=R22(θ−sinθ)

The estimated arch ring face area is the difference between the outer and inner circular-segment areas. The outer segment uses the extrados radius and the same central angle:

Ar=Ro22(θ−sinθ)−Ao

Finally, the estimated masonry volume is the arch ring face area multiplied by the wall depth d:

V=Ard

Calculated outputMathematical basis
Intrados RadiusRadius from span and rise
Extrados RadiusIntrados radius plus ring thickness
Central AngleIncluded angle of the circular arch
Intrados and Extrados LengthRadius multiplied by central angle
Opening AreaInner circular-segment area
Arch Ring Face AreaOuter segment area minus opening area
Masonry VolumeRing face area multiplied by wall depth

Practical Use Cases

Builders can use the calculator to estimate the radius, curved lengths, ring face area, and masonry volume before setting out a circular doorway, window, garden wall, or other curved opening.

Masons can use the intrados and extrados lengths to plan curved brick or stone courses, estimate facing material, and check the effect of changing the arch ring thickness.

Structural engineers can use the geometric results as preliminary dimensions for layout and quantity studies, while using separate structural analysis for loads, stresses, reinforcement, and support conditions.

Homeowners and renovators can compare different span and rise combinations when planning an arched entrance, fireplace opening, partition, or decorative masonry feature and obtain an initial material-volume estimate.

Architects and designers can use the central angle, opening area, and curved surface lengths to coordinate drawings, formwork concepts, glazing layouts, and finish-material requirements.

Accuracy and Limitations

The calculator is mathematically accurate for a circular arch defined by the entered span and rise, with the arch ring treated as a constant radial thickness and the masonry volume treated as face area multiplied by wall depth. It does not analyze structural capacity, thrust, settlement, voussoir joints, mortar allowances, construction tolerances, reinforcement, non-circular curves, skewed arches, or irregular wall geometry. The results are estimates for planning and quantity calculations, and should be checked against detailed drawings and structural requirements before construction.

Quick Examples

Example #1: Standard Brick Arch

Estimate the geometry and masonry requirements for a typical residential brick arch.

Example #2: Garden Gate Arch

Calculate arch dimensions and material volume for a curved garden entrance or pedestrian gate.

Example #3: Large Stone Archway

Estimate the curved lengths, face area, and masonry volume for a substantial stone arch opening.

Frequently Asked Questions

The opening span is the clear horizontal distance between the two spring points where the arch begins. Measure between the inside faces of the opening, not across the surrounding masonry.

Arch rise is the vertical distance from the spring line to the highest point of the inner arch curve. Use the same reference line that connects the two spring points.

Arch ring thickness is the radial distance between the inner and outer curves of the arch. It represents the depth of masonry or structural material surrounding the opening.

The intrados length describes the inner curved face, while the extrados length describes the outer curved face. These measurements help with lining, formwork, masonry layout, and finishing materials.

Estimated masonry volume provides a basis for calculating the amount of material needed for the arch ring across its wall depth. Add an appropriate allowance for joints, cutting, waste, and construction tolerances.

Cite This Calculator