Arch Calculator
Plan your architectural & design project with our free arch calculator. Get precise measurements, material lists, and budgets.
Formula
R = (S²/8H) + H/2 | Arc = R × θ
For a segmental arch, the radius equals the span squared divided by eight times the rise, plus half the rise. The arc length equals the radius times the central angle in radians. For semicircular arches, the radius is simply half the span.
Worked Examples
Example 1: Semicircular Doorway Arch
Problem: A semicircular arch has a span of 3 meters. Calculate the radius, arc length, and area.
Solution: Radius = Span / 2 = 3 / 2 = 1.5 m\nArc Length = π × R = π × 1.5 = 4.7124 m\nArea = (π × R²) / 2 = (π × 2.25) / 2 = 3.5343 m²\nPerimeter = Arc Length + Span = 4.7124 + 3 = 7.7124 m
Result: R = 1.5 m | Arc = 4.7124 m | Area = 3.5343 m²
Example 2: Segmental Window Arch
Problem: A segmental arch above a window has a span of 2 meters and a rise of 0.4 meters. Find the radius and arc length.
Solution: R = (S²)/(8H) + H/2 = (4)/(3.2) + 0.2 = 1.25 + 0.2 = 1.45 m\nHalf angle = arcsin(S/(2R)) = arcsin(1/1.45) = 0.7596 rad\nCentral angle = 2 × 0.7596 = 1.5193 rad\nArc Length = R × θ = 1.45 × 1.5193 = 2.2030 m
Result: R = 1.45 m | Arc Length = 2.2030 m
Frequently Asked Questions
How do you calculate the radius of a segmental arch?
The radius of a segmental arch is calculated using the formula R = (S² / (8H)) + (H / 2), where S is the span (chord length) and H is the rise (sagitta or height at the center). This formula is derived from the geometric relationship between a chord and its corresponding circular segment. For example, if you have a span of 4 meters and a rise of 1 meter, the radius would be R = (16/8) + (1/2) = 2 + 0.5 = 2.5 meters. The radius must always be greater than half the span; otherwise, the arc cannot form. This calculation is essential for laying out arch formwork and determining the curvature needed for brick or stone arches in construction.
What is the difference between an arch and a vault?
An arch is a two-dimensional curved structural element that spans an opening, transferring loads through compression to supports called abutments or piers. A vault is essentially an arch extended in the third dimension — it is a three-dimensional arched structure forming a ceiling or roof. A barrel vault is a semicircular arch extruded along a straight line. A groin vault is formed by the intersection of two barrel vaults at right angles. A ribbed vault uses a framework of arched ribs with lighter infill panels. While arch calculations focus on the cross-sectional geometry (radius, span, rise, arc length), vault calculations additionally require the depth or length dimension to determine surface area, volume, and material quantities for the full three-dimensional structure.
How is arc length calculated for an arch?
Arc length depends on the type of arch and its geometry. For a semicircular arch, the arc length is simply half the circumference of a full circle: L = π × R, where R is the radius (half the span). For a segmental arch, you first calculate the central angle using θ = 2 × arcsin(S / (2R)), where S is the span and R is the radius, then the arc length is L = R × θ (with θ in radians). For a parabolic arch, the arc length requires integration: L = ∫√(1 + (dy/dx)²) dx over the span, which often requires numerical methods. Accurate arc length calculations are essential for estimating material quantities — for example, the number of bricks or stones needed for an arch, the length of reinforcing steel, or the amount of formwork required during construction.
Can I use Arch Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.