To determine the dimensions of a segmental arch (a circular arch that is less than a full semicircle), follow these mathematical steps:

1. Measure the Span and Rise: First, determine the required width (span, denoted as 'W') and height (rise, denoted as 'H') of your arch. These are the two primary inputs for all calculations.
2. Calculate the Radius (R): The radius of the circle that forms the arch is found using the formula:
   R = (H / 2) + (W² / (8 × H))This formula is derived from the intersecting chords theorem in geometry.
3. Calculate the Arc Angle (θ): The angle that the arc subtends at the center of the circle can be found using trigonometry. The formula is:
   θ = 2 × arcsin( (W / 2) / R )The result is typically in radians and should be converted to degrees (multiply by 180/π).
4. Calculate the Arc Length (L): The length of the curved part of the arch is calculated using the radius and the arc angle (in radians).
   L = R × θ (where θ is in radians)

• Span (W): The total width of the arch, measured from the two points where the arch begins.
• Rise (H): The height of the arch from its baseline (the line connecting the start and end points) to its highest point.
• Radius (R): The radius of the complete circle from which the arch's curve is a segment.
• Arc Length (L): The distance along the curved edge of the arch. This is longer than the span.
• Arc Angle (θ): The angle formed by the two radii that connect the center of the circle to the start and end points of the arch.
• Segmental Arch: An arch that is a segment of a circle, i.e., less than a full 180-degree semicircle.

"When laying out an arch for cutting, the most reliable method is to use a trammel or a simple beam compass set to the calculated radius. Find the center point of the arch's baseline (span) and measure down from the arch's peak a distance equal to the radius minus the rise (R - H). This gives you the center of the circle. From there, you can swing a perfect arc." - Master Carpenter

"For structural arches, especially in masonry, the curve's shape is critical for distributing load. While a circular arch is common, for very wide spans, an elliptical or parabolic arch might be structurally superior. Always consult an engineer for load-bearing applications."

Example 1: Arched Bookcase Top
You're building a bookcase that is 36 inches wide (span) and want a decorative arch at the top with a height of 6 inches (rise).
Radius: `(6/2) + (36² / (8 * 6)) = 3 + 1296 / 48 = 30` inches.
You would set a trammel to a 30-inch radius to draw the perfect curve on your material.

Example 2: Garden Gateway Arch
An opening for a garden gate is 60 inches wide, and you want the arch to be 20 inches high.
Radius: `(20/2) + (60² / (8 * 20)) = 10 + 3600 / 160 = 32.5` inches.
Arc Length: `32.5 * (2 * asin((60/2)/32.5)) ≈ 69.8` inches. You'll need a piece of wood or metal at least this long to form the arch.

• Confusing Span and Arc Length: The span is the straight-line width, while the arc length is the curved distance. Material length must be based on the arc length, not the span.
• Incorrect Center Finding: A common error is assuming the circle's center lies on the arch's baseline. For a segmental arch, the center is always located below the baseline.
• Making the Rise Too High: For a given span, the rise cannot be more than half the span for a simple circular arch. A rise equal to half the span creates a perfect semicircle.
• Not Verifying with a Drawing: Always make a scaled drawing or a full-size template on cardboard or hardboard before cutting your final material. This allows you to visually check the proportions and catch any calculation errors.

• Architectural Design: Designing windows, doorways, and vaulted ceilings.
• Custom Woodworking: Creating curved furniture parts like headboards, aprons on tables, or decorative valances.
• Construction and Masonry: Building arched supports for bridges, entryways, or fireplaces.
• Landscape Architecture: Designing arched trellises, arbors, and garden bridges.
• Set and Prop Design: Building scenery for theater or film that requires arched elements.