Arch Calculator
Plan your architectural & design project with our free arch calculator. Get precise measurements, material lists, and budgets.
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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.
Last reviewed: December 2025
Worked Examples
Example 1: Semicircular Doorway Arch
Example 2: Segmental Window Arch
Background & Theory
The Arch Calculator applies the following established principles and formulas. Structural and construction engineering is governed by fundamental load analysis, material science, and regulatory standards that ensure the safety and durability of built structures. The primary distinction in load analysis is between dead loads — the permanent self-weight of structural elements, finishes, and fixed equipment — and live loads, which represent variable occupancy, furniture, and environmental forces such as wind and snow. These are combined using factored load equations, such as the ASCE 7 formula U = 1.2D + 1.6L, where D is dead load and L is live load. Concrete mix design is governed by the water-cement (w/c) ratio, which is the primary determinant of compressive strength and durability. A w/c ratio of 0.40–0.45 typically yields concrete with 28-day compressive strengths of 30–40 MPa. Common mix ratios by weight for structural concrete are approximately 1 part cement : 1.5–2 parts sand : 3 parts coarse aggregate. Structural steel is characterized by its yield strength (the stress at which permanent deformation begins, typically 250–350 MPa for mild steel) and ultimate tensile strength (typically 400–500 MPa). Mid-span deflection of a simply supported beam under a central point load is given by δ = FL³ / (48EI), where F is force, L is span length, E is Young's modulus, and I is the second moment of area. Building insulation is rated by R-value, a measure of thermal resistance in units of m²·K/W (SI) or ft²·°F·h/BTU (imperial). Higher R-values indicate greater resistance to heat flow. Foundation design depends on the allowable bearing capacity of the underlying soil, which ranges from approximately 75 kPa for soft clay to over 10,000 kPa for bedrock. Drainage gradients for surface water are typically specified as a minimum of 1–2% slope away from building foundations to prevent hydrostatic pressure and water infiltration.
History
The history behind the Arch Calculator traces back through the following developments. The history of construction engineering spans thousands of years of accumulated empirical knowledge and, more recently, rigorous scientific analysis. The ancient Egyptians built the Great Pyramid of Giza around 2560 BCE using an estimated 2.3 million stone blocks, demonstrating sophisticated logistics, geometry, and workforce organization. Roman engineers advanced the field dramatically through the use of pozzolanic concrete — a mixture of volcanic ash, lime, and seawater — enabling the construction of the Pantheon dome (43.3 m diameter, completed around 125 CE) and a vast network of aqueducts and roads across the empire. Cast iron emerged as a structural material during the Industrial Revolution, first used prominently in the Iron Bridge at Coalbrookdale, England, completed in 1779. Wrought iron and later steel allowed far greater spans and heights. The Eiffel Tower, completed in 1889, demonstrated the structural possibilities of wrought iron at scale and influenced the development of steel-frame skyscraper construction in Chicago and New York. Reinforced concrete was systematically developed by Joseph Monier, a French gardener, who patented iron-reinforced concrete pots and panels in the 1860s, and later by engineers including François Hennebique who created the first comprehensive reinforced concrete framing system in the 1890s. The 1906 San Francisco earthquake caused widespread devastation and galvanized the engineering profession to develop seismic design provisions. Subsequent earthquakes — including the 1971 San Fernando and 1994 Northridge events — drove successive improvements in seismic codes, base isolation technology, and ductile detailing of reinforced concrete and steel frames. Building codes became increasingly standardized in the twentieth century, with the International Building Code (IBC) first published in 2000 providing a unified model code adopted across much of the United States. Building Information Modeling (BIM) emerged in the 2000s as a digital workflow integrating architectural, structural, and MEP design into a unified three-dimensional model, fundamentally changing coordination practices across the industry.
Frequently Asked Questions
Sources & References
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.
References
Reviewed by Abdullah, Technical Content Specialist · Editorial policy