Circular Sector (Pie Slice) Area and Arc
The pie slice between two radii, and its arc - a curved patio or bed, a sprinkler coverage wedge, a gear or cam sector.
Example
You enter
- Radius r 5
- Central angle (deg) 60
You get
- Sector area 13.09 (square units)
- Arc length 5.236
- Chord 5
Checked against published worked examples from: Naval Education and Training Professional Development and Technology Center (U.S. Navy); Mathwords (Bruce Simmons).
Details, formula, and sources
ScopeThe pie slice between two radii, and its arc - a curved patio or bed, a sprinkler coverage wedge, a gear or cam sector, or a fan of pavers. area = (1/2) r^2 theta = (angle/360) pi r^2, arc = r theta, chord = 2 r sin(theta/2). A 5 ft radius, 60-degree sector covers 13.1 ft^2 with a 5.24 ft curved edge and a 5.0 ft chord across the mouth (perimeter 15.24 ft); open it to 360 degrees for the full circle, 78.5 ft^2. The circular segment (the chord-and-arc region) and the cylindrical wedge are separate shapes. A shop and layout aid; verify critical dimensions on the work.
Formula or tabletheta = angle x pi/180; area = (1/2) r^2 theta = (angle/360) pi r^2; arc = r theta; chord = 2 r sin(theta/2); perimeter = arc + 2 r.
Source and editionThe circular sector area (1/2) r^2 theta and arc length r theta - standard plane geometry as in Machinery's Handbook (Industrial Press), by name; public domain.
Free accessPure plane geometry, public; the radius and central angle are user-supplied.
What governsEstimate. AHJ and licensed professional govern.
API field namesradius, angle_deg, area, arc_length, chord
- Sector area/arc area = (1/2) r^2 theta, arc = r theta; the fraction angle/360 of the circleplane geometry
- Scope sector (two radii + arc); the segment (chord + arc) is separatescope of this tile