Arches & curves
Arc length & chord
- where
- arc length (m)
- radius (m)
- included angle (degrees)
- chord (m)
- sector area (m²)
Explain this equation
This gives the arc length, chord and sector area of a circle from its radius and included angle.
R is the radius of the line you are setting out, in metres. θ is the included angle at the centre in degrees. a is the distance along the curve, c is the straight-line distance between its ends, and A is the area of the pie-slice between the curve and the centre.
An arc is R times its angle in radians, and multiplying degrees by π/180 converts them to radians. The chord comes from splitting the sector into two right-angled triangles, each with a half-angle of θ/2, giving 2R sin(θ/2). The sector area is the same fraction of the whole circle, πR², as θ is of 360°.
A 2 m radius over 90° gives a 3.14 m arc, a 2.83 m chord and a 3.14 m² sector. The arc is always a little longer than the chord.
Watch out: the angle is limited to 180°. Use the radius of the line you are actually building to, such as the face or centreline.
Calculate a circular arc from radius and included angle.
Enter your values to see the result.
Good to know
Use the radius of the intended setting-out line. Limited to 0–180 degrees for the minor arc.
Sources
- Geometry / arithmetic
Estimates are not design approval or British Standards compliance checks. Structural arch, retaining-wall, wind, foundation, lintel and masonry-capacity design are not performed. Use the project engineer’s specification.