Sector Area Calculator

Calculate sector area, arc length, and chord length of a circle. Perfect for geometry students, teachers, and professionals.

Sector Area Formula: Area = (θ/360°) × π × r² (when θ is in degrees)

or Area = (1/2) × r² × θ (when θ is in radians)

Radius of the circle
Central angle of the sector
Unit of measurement for the angle
Select what to calculate
r=5, θ=90°
r=10, θ=60°
r=8, θ=120°
r=6, θ=π/3 rad
r=12, θ=π rad
r=15, θ=45°
Calculating...

Understanding Circular Sectors

A circular sector is a portion of a circle enclosed by two radii and an arc. It's like a "pie slice" of a circle. The size of the sector is determined by the central angle θ and the radius r of the circle.

Key Components:

  • Radius (r): Distance from the center to any point on the circle
  • Central Angle (θ): Angle formed at the center by the two radii
  • Arc: The curved portion of the sector's boundary
  • Chord: The straight line connecting the two endpoints of the arc

Sector Formulas

Property Formula (Degrees) Formula (Radians) Example
Sector Area A = (θ/360°) × πr² A = (1/2) × r² × θ r=5, θ=60° → A=13.09
Arc Length L = (θ/360°) × 2πr L = r × θ r=5, θ=60° → L=5.24
Chord Length c = 2 × r × sin(θ/2) r=5, θ=60° → c=5
Perimeter P = 2r + L P = 2r + L r=5, L=5.24 → P=15.24
Full Circle Area A = πr² r=5 → A=78.54
Circumference C = 2πr r=5 → C=31.42

Angle Conversion

1

Degrees to Radians: Multiply degrees by π/180

Radians = Degrees × (π/180)

2

Radians to Degrees: Multiply radians by 180/π

Degrees = Radians × (180/π)

3

Common Conversions:

360° = 2π rad, 180° = π rad, 90° = π/2 rad,
60° = π/3 rad, 45° = π/4 rad, 30° = π/6 rad

Applications of Sector Calculations

  • Engineering: Designing gears, pulleys, and curved structures
  • Architecture: Calculating materials for circular designs and arches
  • Physics: Analyzing rotational motion and angular displacement
  • Land Surveying: Calculating areas of circular plots and parcels
  • Manufacturing: Cutting materials with circular patterns
  • Computer Graphics: Rendering circular UI elements and animations

Calculator Features:

  • Calculates all sector properties: area, arc length, chord length, and perimeter
  • Works with both degrees and radians
  • Provides visual representation of the sector
  • Shows step-by-step calculations and formulas used
  • Includes common examples for quick testing

Frequently Asked Questions

A sector is the region bounded by two radii and an arc (like a pie slice). A segment is the region bounded by a chord and the arc subtended by that chord. Essentially, a segment is a sector minus the triangle formed by the two radii and the chord.

If you know the arc length (L) and radius (r), you can find the angle first: θ = L/r (in radians) or θ = (L × 360°)/(2πr) (in degrees). Then use the sector area formula with this angle.

The maximum area occurs when the central angle is 360° (or 2π radians), which gives the full circle area: A = πr². So a sector can never have an area greater than the area of the entire circle.

Yes, for angles greater than 60° (π/3 radians), the chord length is greater than the radius. The chord length equals the radius when θ = 60°, and for larger angles, the chord continues to increase until it reaches the maximum of 2r (the diameter) when θ = 180°.

The perimeter of a sector is the sum of the lengths of the two radii and the arc: P = 2r + L, where L is the arc length. This gives the total distance around the sector.