Arch Calculator
Calculate arch dimensions, rise, span, and radius for construction and masonry projects.
What do these terms mean?
Span — the total horizontal width of the arch opening (chord length).
Rise — the vertical height from the springline (base) to the arch apex (sagitta).
Radius — the radius of the circle that forms the arch.
What This Arch Calculator Does
This calculator determines the key geometric dimensions of a segmental arch: the rise, span, and radius. Given any two of these values, it computes the third, enabling accurate layout and material estimation for construction and masonry projects.
The tool is intended for arches with a circular arc profile, commonly used in doorways, windows, bridges, and decorative masonry. It eliminates manual trigonometric calculation and reduces the risk of measurement error on site.
How the Calculation Works
The calculator uses the geometric relationship between the span (the horizontal distance between supports), the rise (the vertical height from the spring line to the crown), and the radius of the circular arc. The underlying formula is derived from the chord and sagitta relationship:
R = (S² / 8Rise) + (Rise / 2)
Where:
- R = radius of the arc
- S = span (chord length)
- Rise = rise (sagitta height)
If you provide the span and rise, the calculator returns the radius. If you provide the span and radius, it returns the rise. If you provide the rise and radius, it returns the span. All values are assumed to be in the same unit of measurement.
How to Use the Calculator
- Enter any two of the three dimensions: span, rise, or radius.
- Ensure all inputs use the same unit (e.g., inches, feet, centimeters).
- The calculator will compute the missing dimension instantly.
No additional configuration is required. The result updates automatically as you adjust the inputs.
Practical Example
Suppose you are building a masonry arch with a span of 48 inches and a desired rise of 6 inches. Enter these two values into the calculator. The tool returns a radius of approximately 51 inches. This radius can then be used to lay out the arc on a template or to calculate the length of the arch's intrados.
If instead you know the span is 48 inches and the radius is 60 inches, the calculator will return a rise of about 4.9 inches. This allows you to verify whether the arch profile meets your design requirements before cutting materials.
Understanding the Results
The computed dimension is the exact geometric value based on the inputs provided. In practice, construction tolerances may apply. For masonry work, slight adjustments are often made for mortar joints and material thickness. The calculator gives you the theoretical centerline or intrados dimension, which you can then adapt to your specific build method.
If the result seems unexpected, double-check that all inputs are in the same unit and that the values are physically plausible. For example, the rise cannot exceed the radius, and the span cannot be greater than twice the radius.
Common Mistakes to Avoid
- Mixing units: Entering span in inches and rise in feet will produce an incorrect result. Always use the same unit for all inputs.
- Confusing rise with height: The rise is measured from the spring line (the horizontal line connecting the arch's supports) to the crown, not from the ground or floor.
- Assuming a semicircular arch: This calculator is for segmental arches (less than a full half-circle). For a true semicircular arch, the rise equals half the span, and the radius equals half the span.
Limitations
This calculator assumes a perfect circular arc. It does not account for elliptical, pointed, or parabolic arch profiles. For non-circular geometries, different formulas are required. Additionally, the tool provides geometric dimensions only; it does not calculate material quantities, load-bearing capacity, or structural suitability.
Practical Use Cases
- Masonry arch construction: Determine the radius for cutting brick or stone voussoirs.
- Door and window framing: Calculate the rise needed for a given span and radius to fit an arched opening.
- Bridge design: Verify arch geometry during preliminary layout.
- Restoration work: Replicate an existing arch by measuring its span and rise to find the original radius.
FAQ
What is the difference between span and radius?
The span is the straight-line distance between the two supports of the arch. The radius is the distance from the center of the circular arc to any point on the curve. For a segmental arch, the radius is always larger than half the span.
Can I use this calculator for a semicircular arch?
Yes, but for a true semicircular arch the rise is exactly half the span, and the radius is exactly half the span. The calculator will work correctly, but you may find it simpler to use those direct relationships.
What unit should I use?
Any unit is fine as long as all inputs are in the same unit. The result will be in that same unit. Common choices are inches, feet, centimeters, or meters.
Why is my result showing a very large or very small number?
This usually indicates a unit mismatch or an input that is physically impossible. For example, a rise larger than the radius or a span larger than twice the radius will produce an invalid geometry. Check your inputs and ensure they are realistic for your project.
Does this calculator account for mortar joints or material thickness?
No. The calculator returns the pure geometric dimension. In masonry work, you may need to adjust for mortar joint thickness or the difference between the intrados and extrados. Use the result as a starting point for your layout template.