Calculate the perimeter of a circular sector from the radius and central angle. Find arc length, sector area, chord length, and segment area with visual breakdowns.
A circular sector is the "pie slice" region bounded by two radii and the arc between them. Its perimeter — the total length around its boundary — consists of two straight edges (the radii) plus the curved arc. The formula is straightforward: P = 2r + rθ, where r is the radius and θ is the central angle in radians.
Sector calculations appear constantly in geometry, engineering, and everyday life. Calculating the length of a pizza crust, the fencing needed around a circular garden section, or the material for a fan blade all reduce to finding a sector's perimeter or area. The arc length alone is L = rθ, and the sector area is A = ½r²θ.
This calculator accepts the radius and the central angle in either degrees or radians, then instantly computes the full perimeter, arc length, area, chord length (the straight-line distance between the arc's endpoints), and the segment area (the region between the chord and the arc). It also shows what fraction of the full circle the sector represents.
Visual bars break down the perimeter into its arc and radius components and compare the sector area to the triangle and segment areas. A reference table provides symbolic formulas for common angles (30°, 45°, 60°, 90°, etc.). Eight presets let you jump to popular configurations instantly. Whether you are studying for a math exam or working on a design project, this tool covers every sector-related measurement you might need.
Perimeter of a Sector problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter radius, central angle, length unit label, and it returns sector perimeter, arc length, sector area, chord length in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Perimeter = 2r + rθ (θ in radians). Arc length = rθ. Sector area = ½r²θ. Chord length = 2r sin(θ/2). Convert degrees to radians: θ_rad = θ_deg × π/180.
Result: Perimeter ≈ 35.71 units
θ = 90° = π/2 ≈ 1.5708 rad. Arc = 10 × 1.5708 ≈ 15.71. Perimeter = 2(10) + 15.71 = 35.71. Area = ½ × 100 × 1.5708 ≈ 78.54.
This perimeter of a sector tool links the entered values (radius, central angle, length unit label, show reference table) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Perimeter of a Sector shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (sector perimeter, arc length, sector area, chord length) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
It is the total distance around the sector: two radii plus the arc length. P = 2r + rθ.
A sector is bounded by two radii and an arc ("pie slice"). A segment is bounded by a chord and the arc above it.
Yes. A sector can have any central angle from 0° up to 360°. Angles above 180° produce "reflex" sectors.
θ = arc length / radius (in radians). Convert to degrees by multiplying by 180/π.
Use consistent length units (cm, m, in, etc.). The angle must be in radians for the formulas; this calculator converts automatically.
The chord is the straight-line distance between the arc endpoints — useful for constructing physical shapes or measuring spans. Use this as a practical reminder before finalizing the result.