Arch Math: Layout Geometry and Joint Calculations

Why Arch Math Matters

An arch transfers load outward and downward to its supports through compression. Every brick (called a voussoir) must be cut or laid so that its bed joints, when extended, all converge at the same center point. If your geometry is off, joints open unevenly, the arch rocks, or mortar is stressed in tension — something masonry cannot tolerate. Getting the numbers right on paper before touching a brick saves material, time, and prevents structural failure.

Key Arch Geometry Terms

Span is the clear horizontal distance between supports (the opening width). Rise is the vertical distance from the spring line to the soffit at the crown. The spring line is the horizontal line where the arch begins to curve upward from its supports. The intrados is the inner curved surface (the underside you see when you look up). The extrados is the outer curved surface. The skewback is the angled cut or course at the base of the arch where voussoirs begin.

Semicircular Arch Layout

For a semicircular arch, the rise equals exactly half the span, so the radius equals half the span as well. The center point sits on the spring line at the midpoint of the opening.

• Radius (R) = Span ÷ 2

To locate the center on graph paper, mark the two support points. Measure the span, find the midpoint, and mark it. That midpoint on the spring line is your center. Swing the radius from that center to define the intrados curve.

Divide the semicircle into an odd number of equal angular divisions so that one voussoir falls exactly at the crown (the top). An odd count — typically 7, 9, 11, or more depending on span — ensures symmetry and a single keystone.

The skewback angle for a semicircular arch at the spring line is always 90° measured from vertical, because the first bed joint must point directly at the center. For segmental arches, this angle varies and must be calculated.

Segmental Arch Layout

A segmental arch is a segment of a circle whose rise is less than half the span. You must calculate the radius from span and rise using the following formula derived from the geometry of a chord and its perpendicular bisector:

R = (4 × Rise² + Span²) ÷ (8 × Rise)

The center point lies below the spring line for a segmental arch. To find it, measure down from the midpoint of the spring line by a distance equal to (R − Rise).

The skewback angle (measured from horizontal at the spring line) equals the angle whose sine is (Half-Span ÷ R), computed as:

Skewback Angle = arcsin(Half-Span ÷ R)

Use your calculator's inverse-sine function (sin⁻¹). Round to the nearest whole degree for layout purposes unless the specification requires more precision.

Voussoir Count and Joint Calculations

Once you know the arc length, you can calculate the number of voussoirs. The arc length of a semicircular arch is:

Arc Length = π × R (half the circumference of a full circle)

For a segmental arch, arc length equals R × central angle in radians, where the central angle in radians = 2 × arcsin(Half-Span ÷ R) converted to radians (multiply degrees by π/180).

Each voussoir occupies the brick unit length plus one mortar joint on the intrados. Because joints taper, the intrados joint is narrower than the extrados joint. Standard practice limits the minimum joint width at the intrados to ¼ inch and the maximum joint width at the extrados to ¾ inch for most brick arches (verify with your project specification or the Brick Industry Association technical notes).

Joint taper per voussoir depends on the radial depth of the arch ring (extrados radius minus intrados radius). The formula is:

Taper = (Extrados Circumference − Intrados Circumference) ÷ Number of Joints

Or equivalently: Taper = (Depth of Arch Ring × 2π × Central Angle in degrees) ÷ (360 × Number of Joints)

If your taper calculation produces an intrados joint below ¼ inch, you must either add more voussoirs, increase the arch ring depth, or specify a purpose-made tapered brick unit (a rubbed or gauged arch unit).

Reading Dimensions Accurately

All layout measurements should be made to the nearest 1/16 inch. When using an architect's scale or framing square, read from the zero mark, hold the tool flat, and read at eye level to avoid parallax error. When using a construction calculator, confirm your unit settings (feet-inch-fractions vs. decimal inches) before entering values.

Safety Notes

Layout work is low-hazard, but the following apply:

• Graph paper and pencil work at a desk: keep the work area clear; a framing square has sharp corners — handle it with care and store it flat so it does not bend or fall.
• Construction calculator: electronic devices near dusty or wet masonry conditions can be damaged; keep it in a protective case when not in use.
• When you move from paper calculations to the job site to transfer layout marks, wear required PPE (hard hat, safety glasses, work boots with slip-resistant soles) because you will be working around masonry operations above.
• Never scale measurements from a printed drawing unless the drawing states it is "drawn to scale" and you have confirmed the print has not been reduced — always use the dimensioned numbers.