Free Arc Calculator — Curved Wall & Segment Dimensions

How to Use the Arc Calculator

How to Calculate Arc Dimensions

An arc is defined by five interrelated dimensions: radius, chord length, height (sagitta), central angle, and arc length. If you know any two of these, this calculator solves for the remaining three. The fundamental relationships come from circle geometry: the chord connects two points on the circle, the sagitta (height) is the perpendicular distance from the chord midpoint to the arc, and the central angle is the angle subtended by the arc at the center of the circle.

Key Formulas

• Chord from radius and angle: Chord = 2 × R × sin(θ/2)
• Height from radius and angle: Height = R × (1 − cos(θ/2))
• Arc length from radius and angle: Arc Length = R × θ (radians)
• Radius from chord and height: R = (C²)/(8H) + H/2
• Segment area: A = (R²/2)(θ − sinθ)
• Sector area: A = (R²/2)(θ)

Common Construction Applications

Arcs appear throughout construction and woodworking. Arched doorways and windows need precise radius and height measurements to frame the curved header. Curved walls in commercial and residential buildings require the radius to calculate stud spacing along the arc. Barrel-vault ceilings use arc dimensions for rib spacing and drywall bending calculations. In cabinetmaking, curved moldings and trim pieces need the chord and height to set up a router jig or bandsaw cut. Even concrete form work for curved retaining walls depends on accurate arc calculations to bend plywood or steel forms.

Measuring Arcs in the Field

The easiest measurements to take on an existing arc are the chord and height. Stretch a string or straight edge between the two endpoints of the arc to measure the chord. Then measure perpendicularly from the midpoint of the string to the highest point of the arc to get the height (sagitta). With these two measurements, the calculator derives the radius, angle, and arc length. For large arcs like curved walls, use a laser distance meter for accuracy. For small arcs in woodworking, a flexible ruler or contour gauge captures the profile precisely.

Pro Tips

• For curved wall framing: Once you know the radius, space studs at standard on-center intervals measured along the chord, not the arc. The difference is negligible for large-radius curves and ensures your sheathing attachment points align.
• Bending plywood for forms: Standard 1/4" plywood bends to a minimum radius of about 24 inches without kerfing. For tighter radii, use bendable (wacky wood) plywood or kerf-cut the back face.
• Check your work: After calculating, verify by confirming that the height is always less than or equal to the radius (height equals radius only for a semicircle at 180°).