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Brick Arch Calculator

Design a segmental or semicircular brick arch: radius from span and rise, arch angle, intrados and extrados lengths, number of bricks per ring (odd, for a keystone) and the tapered joint widths.

Span (opening width)

Rise

Brick thickness on the arch face

Joint at the intrados

Ring depth (brick length)

Number of rings

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How it works

R = (span²/4 + rise²) ÷ (2 × rise) · arch angle θ = 2 × asin(span ÷ 2R) · intrados = R × θ · bricks per ring = intrados ÷ (brick + joint), rounded to an odd number · extrados = (R + ring depth) × θ

A segmental arch is part of a circle through the two springing points and the crown, so its radius follows from the span and rise by the intersecting-chords theorem; a rise of half the span gives a semicircle. Bricks (voussoirs) are counted along the inner curve (the intrados) at one brick plus one joint each, rounded to an odd number so a single keystone sits at the crown. Because the outer curve is longer, rectangular bricks leave tapered joints that are wider at the extrados; joints much over about 2.5 cm suggest cutting tapered voussoirs or using more, thinner rings. Each extra ring starts one ring depth and joint further out and needs more bricks.

Worked example

A 120 cm span with a 30 cm rise: R = (3,600 + 900) ÷ 60 = 75 cm; θ = 2 × asin(60 ÷ 75) = 106.3°. The intrados is 75 × 1.855 = 139.1 cm, so 139.1 ÷ (6.5 + 1) = 18.5 → 19 bricks (odd, for the keystone). The joints are 0.82 cm inside and, with a 23 cm ring (extrados 181.7 cm), 3.07 cm outside.

Sources: Brick Industry Association — Technical Note 31: Brick Masonry Arches · Circular segment geometry (Wolfram MathWorld)

Frequently asked questions

How do I find the radius of an arch from its span and rise?

R = (span²/4 + rise²) ÷ (2 × rise). For a 120 cm span and 30 cm rise that is 75 cm. Strike the curve with a string of that length from a point R − rise below the crown.

Why should an arch have an odd number of bricks?

An odd count puts one brick — the keystone — exactly at the crown, so the arch is symmetrical and the last brick wedges both halves tight.

What rise should a segmental arch have?

Traditional segmental arches use a rise of about 1/8 to 1/4 of the span; flatter arches push harder sideways on the abutments. A semicircular arch has a rise of half the span.

How wide can the joints be?

Keep intrados joints near 6–10 mm. If the outer joints exceed about 25 mm, use two thinner rings (bricks on edge) or cut tapered voussoirs.

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This calculator is for educational purposes. Double-check important results before acting on them.