What is the difference between a sector and a segment of a circle?

A circle sector is shaped like a slice of pie, bounded by two radii from the center and an arc on the perimeter. A circle segment is bounded by a straight line (chord) connecting two points on the circumference and the arc between them, without reaching the center.

How do you convert degrees to radians for sector calculations?

Multiply the angle in degrees by π and divide by 180: Radians = Degrees × (π / 180). For example, 90° = 90 × (π/180) = π/2 ≈ 1.5708 radians.

What is the sagitta of an arc?

The sagitta (Latin for "arrow") is the height of an arc. It is the perpendicular distance measured from the midpoint of a chord straight up to the highest point of the curved circular arc.

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Circle Sector Calculator

Calculate arc length, sector area, circular segment area, chord length, sagitta (arch height), and sector perimeter for any circle radius and central angle.

Interactive Circle Sector Calculator Workspace

Live Solver

Circle Sector & Segment Calculation: Compute arc length, sector area, chord length, sagitta (arch height), and circular segment area in degrees or radians.

Circle Radius (r)

Central Angle (θ)

Sector & Segment Outputs

Sector Area18.8496

Arc Length (L)6.2832

Segment Area3.2611

Chord Length (c)6.0000

Sagitta (Height)0.8038

Sector Perimeter18.2832

Formula Derivations (θ = 60.00° = 1.0472 rad)

1. Arc Length (L): L = r × θ = 6 × 1.0472 = 6.2832

2. Sector Area: A_sector = (1/2) × r² × θ = 0.5 × 36 × 1.0472 = 18.8496

3. Chord Length: Straight line connecting arc endpoints = c = 2 × r × sin(θ/2) = 6.0000

4. Circular Segment: Enclosed space between chord and arc: A_segment = A_sector - A_triangle = 18.8496 - 15.5885 = 3.2611

Mathematical Formula & Variables

Equation ModelArc Length = r × θ, Sector Area = (1/2) × r² × θ, Segment Area = (1/2) × r² × (θ - sin θ)

Analyzes sub-regions of a circle bounded by two radii and an arc (sector) or bounded by an arc and straight chord (segment), with angle in radians or degrees.

Variable Definitions

Symbol	Description
r (Radius)	Distance from circle center to outer perimeter arc.
θ (Central Angle)	Angle subtended at circle center in degrees or radians.
L (Arc Length)	Curved boundary distance along the circle circumference: L = r × θ_rad.
c (Chord Length)	Straight line distance between arc endpoints: c = 2r × sin(θ/2).
h (Sagitta)	Perpendicular height of the arch between chord midpoint and arc: h = r(1 - cos(θ/2)).

How to Use the Circle Sector Calculator

Enter the circle Radius (r).
Select your angle measurement unit: Degrees (°) or Radians (rad).
Input the Central Angle (θ) or use one of the quick preset angle buttons.
Review Sector Area, Arc Length, Chord Length, Sagitta, and Circular Segment Area.

Practical Example Calculation

Scenario Context: A civil engineer is designing a curved roadway segment with a curve radius of 50 meters and a central deflection angle of 60 degrees.

Step 1: Identify given values: Radius (r) = 50 m, Angle (θ) = 60°.

Step 2: Convert angle to radians: θ_rad = 60° × (π / 180°) = π/3 ≈ 1.0472 radians.

Step 3: Calculate Arc Length (L): L = r × θ_rad = 50 × 1.0472 ≈ 52.36 meters.

Step 4: Calculate Sector Area: A_sector = 0.5 × r² × θ_rad = 0.5 × 2500 × 1.0472 ≈ 1,309.00 square meters.

Step 5: Calculate Chord Length (c): c = 2r × sin(θ/2) = 100 × sin(30°) = 100 × 0.5 = 50.00 meters.

Step 6: Calculate Segment Area: A_segment = A_sector - (0.5 × r² × sin(θ)) = 1309.00 - (0.5 × 2500 × 0.8660) = 1309.00 - 1082.53 = 226.47 square meters.

Step 7: Calculate Sagitta (arch height): h = r × (1 - cos(30°)) = 50 × (1 - 0.8660) = 6.70 meters.

The curved road sector has an arc length of 52.36 m, chord length of 50.00 m, sagitta of 6.70 m, and sector area of 1,309.00 m².

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