Arch Calculator (radius, arc length & angle from span and rise)

Understanding the arch calculator — concept and principle

This calculator finds the radius, arc length, central angle and area of a round (circular) arch from its span and rise. Builders, woodworkers and garden designers use it to set out an arch for a doorway, window, bridge or trellis.

Span and rise → radius: with the default inputs the result is R = 75 cm (arc 139.1 cm · 106.3°).

How it is calculated

Radius R = (span² ÷ 4 + rise²) ÷ (2 × rise)

Central angle θ = 2 × atan2(span ÷ 2, R − rise)

Arc length = R × θ (θ in radians)

Area under the arch = R² × (θ − sin θ) ÷ 2

Centre below the springing line = R − rise

When the rise is half the span the arch is a semicircle; a lower rise gives a segmental arch.

The principle

A round arch is part of a circle passing through the two springing points and the crown. Draw a line from the centre to a springing point: it forms a right triangle with half the span and R − rise, so R² = (span/2)² + (R − rise)². Solving for R gives the formula. The angle and arc length follow from basic circle geometry, and the area is a circular segment.

Worked example

Default inputs: span 120 cm, rise 30 cm.

R = (120² ÷ 4 + 30²) ÷ (2 × 30) = (3,600 + 900) ÷ 60 = 75 cm.
θ = 2 × atan2(60, 45) = 2 × 53.13° = 106.3°; rise-to-span 0.25.
Arc length = 75 × 1.8546 = 139.1 cm.
Area = 75² × (1.8546 − 0.96) ÷ 2 = 2,516 cm²; centre 45 cm below the springing line → segmental.

Each point is the calculator re-run with only “Span, width of the opening (cm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Span, width of the opening (cm)	Radius (cm)
30	18.8
60	30
90	48.8
120	75
150	108.8
180	150
240	255

How to read the result

Use the radius to set out the curve with a trammel or string line from the centre point. The arc length tells you how long a flexible strip or trim must be. For masonry or structural arches, shape alone does not guarantee strength; thrust at the supports must be resisted.

Common mistakes

Measuring rise from the wrong baseline; it is from the springing line to the crown.
Confusing a segmental arch with an elliptical one, which has no single radius.
Using inside measurements for one dimension and outside for another.

Keep in mind: This gives geometry only. Structural arches and openings in load-bearing walls need local building codes and a qualified professional.

Frequently asked questions

What is a segmental arch?

An arch that is less than a half circle, so its centre lies below the springing line.

How do I mark the curve?

Fix a string or trammel at the centre point, set it to the radius and draw the curve.

Can I use this for an elliptical arch?

No. Elliptical arches have changing curvature and need different formulas.

Sources: Zwillinger D (ed.), CRC Standard Mathematical Tables and Formulas, 33rd ed., 2018, §4 Geometry

Checked 9 Nov 2026. Circle through three points and circular-segment formulas (radius from chord and sagitta, arc length R·θ, segment area R²(θ − sin θ)/2): D. Zwillinger (ed.), CRC Standard Mathematical Tables and Formulas, 33rd ed. (2018), §4 (Geometry).

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