Sector Perimeter Calculator

A sector is a "pie slice" shaped portion of a circle, bounded by two radii (straight lines from the center to the edge) and an arc (the curved portion of the circumference between them). The central angle θ determines the size of the sector. A 90° sector is a quarter of the circle (quadrant), a 180° sector is half the circle (semicircle), and a 360° sector is the complete circle. Sectors are fundamental in geometry, appearing in pie charts, pizza slices, camera apertures, and many engineering applications.

The perimeter of a sector consists of three parts: two radii and one arc. The formula is P = 2r + s, where r is the radius and s is the arc length. The arc length is calculated as s = rθ (when θ is in radians) or s = (θ/360) × 2πr (when θ is in degrees). Combined: P = 2r + rθ = r(2 + θ) for radians, or P = 2r + (θ/360) × 2πr for degrees. Example: For r = 10 cm and θ = 90°: Arc = (90/360) × 2π × 10 = 15.71 cm, Perimeter = 2(10) + 15.71 = 35.71 cm.

Arc length is the distance along the curved part of the sector (following the circumference), while chord length is the straight-line distance between the two endpoints of the arc. Arc length: s = rθ (radians) or s = (θ/360) × 2πr (degrees). Chord length: c = 2r × sin(θ/2). For the same angle, arc length is always greater than chord length (except when θ = 0). The difference becomes more pronounced as the angle increases. For a semicircle (180°), the chord equals the diameter (2r), while the arc equals half the circumference (πr).

Degrees and radians are two ways to measure angles. A full circle is 360° or 2π radians. Conversion formulas: Degrees to radians: radians = degrees × (π/180). Radians to degrees: degrees = radians × (180/π). Common values: 30° = π/6 rad ≈ 0.524 rad; 45° = π/4 rad ≈ 0.785 rad; 60° = π/3 rad ≈ 1.047 rad; 90° = π/2 rad ≈ 1.571 rad; 180° = π rad ≈ 3.142 rad; 360° = 2π rad ≈ 6.283 rad. Radians are often preferred in mathematics because they simplify many formulas.

The area of a sector is a fraction of the total circle area. Formula: A = (1/2)r²θ (when θ is in radians) or A = (θ/360) × πr² (when θ is in degrees). This makes sense because a sector is θ/360 of the full circle. Example: For r = 10 cm and θ = 90° (quarter circle): A = (90/360) × π × 10² = (1/4) × π × 100 = 78.54 cm². The area is proportional to the central angle—double the angle, double the area.

A segment is the region between a chord and its arc, while a sector is the region between two radii and an arc (the "pie slice"). To find segment area, subtract the triangle area from the sector area: Segment Area = Sector Area - Triangle Area = (1/2)r²θ - (1/2)r²sin(θ) = (1/2)r²(θ - sin(θ)), where θ is in radians. For small angles, the segment is very thin. For larger angles, the segment becomes more significant. At 180° (semicircle), the chord is the diameter, and the segment equals the sector (half the circle).

Various sector sizes have special names: Dodecant (30°): 1/12 of a circle. Octant (45°): 1/8 of a circle. Sextant (60°): 1/6 of a circle, historically important for navigation. Quadrant (90°): 1/4 of a circle, a quarter turn or right angle. Third (120°): 1/3 of a circle. Semicircle (180°): 1/2 of a circle, a straight angle. These named fractions appear frequently in engineering, navigation, astronomy, and everyday measurements. The sextant navigation instrument, for example, measures angles up to 60°.

Sectors appear in many real-world applications: (1) Pie charts - Each slice is a sector showing proportional data. (2) Pizza slices - A pizza cut into 8 equal pieces creates 45° sectors. (3) Clock faces - Each hour marks a 30° sector. (4) Camera aperture - Controls light using sector-shaped openings. (5) Windshield wipers - Sweep across a sector-shaped area. (6) Radar displays - Show circular sectors of coverage. (7) Architecture - Sector windows, domes, and arches. (8) Sports - Hockey goal crease, baseball outfield sections. (9) Surveying - Angular measurements of land parcels.