Calculate the radius, arc length, and central angle for circular arches in construction and architectural design.
2026-04-09T00:00:00Z
A circular arch is a curved structural element that spans an opening by distributing weight along a curved path. Architecturally, it's a segment of a circle defined by its span (the horizontal distance across the opening) and its rise (the vertical height from the spring line to the highest point of the curve).
To construct an arch, you need to know the radius of the circle that generates the curve. This radius is calculated based on the span and rise dimensions. Once you have the radius, you can create a template or jig that guides the cutting and placement of radial members (stones, wood, metal) that build the arch structure. Arches are found in architecture, bridge construction, doorways, windows, and decorative applications throughout history and modern construction.
Where R is the radius of the arch circle. This formula comes from the geometric relationship between a chord (the span), its sagitta (the rise), and the radius of the circle. The radius determines the curvature of the arch and is essential for layout and construction.
The central angle is the angle subtended by the arch at the center of the circle. This angle determines how the radial members should be cut and arranged. Convert to degrees by multiplying by (180° ÷ π).
The arc length represents the actual curved distance along the arch. This is useful for determining material quantities needed to build the arch, such as bricks, stones, or curved molding.
Designing an archway for a 60-inch-wide doorway with 12-inch rise:
Measure down from the spring line (the base of the arch) by exactly the radius distance. This point becomes your compass center for drawing or trammel bar for constructing the arch.
The spring line is the imaginary horizontal line where the arch begins to curve upward from its base. It's the point where the vertical and curved portions of the structure meet.
Yes. For a perfect semi-circle, the rise equals exactly half the span. The radius will also equal half the span. For example, a 60" span semi-circular arch has a rise of 30" and radius of 30".
Rise is the vertical distance from the spring line to the crown (highest point) of the arch. Height usually refers to total vertical dimension of the opening, including any structural elements.
Create a template or jig set to the calculated radius. Radial members (stones, wood segments, metal pieces) are cut so their edges align with this radius and point toward the center.
Switch to Metric mode and enter your span and rise in centimeters. The calculator will return all results in centimeters. All formulas work identically regardless of unit system.
This calculator returns geometric properties only. Structural capacity depends on material, support conditions, load paths, and safety factors; always consult a structural engineer for load-bearing arches.
No. This tool assumes circular-arc geometry. Elliptical or parabolic arches use different equations and templates; use specialized geometry tools or CAD for those forms.
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