Circle Sector Calculator | Arc, Area & Chord
Calculate arc length, sector area, chord length, segment area, and perimeter of a circle sector. Supports 4 solve-for modes (r+θ, arc+θ, area+θ, r+arc), degrees or radians, 7 length units, SVG diagram, presets, and step-by-step working.
Solve for (given known values)
Angle unit
Length unit
Presets
Press Enter to calculate · Esc to reset
What Is the Circle Sector Calculator | Arc, Area & Chord?
A circle sector is the region bounded by two radii and the arc between them, the familiar pie-slice shape. Its geometry is completely determined by two values: the radius r and the central angle θ. This calculator computes all five derived quantities (arc length, sector area, chord length, segment area, perimeter) and also supports four solve-for modes so you can find the missing value when only two of the five are known.
- ›Four input modes, start from r+θ, arc+θ, area+θ, or r+arc. The calculator derives the rest.
- ›Degrees or radians, switch freely; all internal math uses radians automatically.
- ›7 length units, mm, cm, m, km, in, ft, yd. Results display in your chosen unit.
- ›SVG diagram, visual reference showing the sector, chord, radii, and arc label, updating with each calculation.
- ›Step-by-step working, expand to see every arithmetic step from input to each output value.
- ›5 presets, clock hand, pizza slice, quarter circle, semicircle, and full circle for quick exploration.
Formula
Arc Length
L = r × θ (θ must be in radians)
Sector Area
A = ½ × r² × θ (θ in radians)
Chord Length
c = 2r × sin(θ / 2)
Segment Area (arc region minus triangle)
S = ½r²(θ − sin θ)
Perimeter (sector boundary)
P = L + 2r = rθ + 2r
| Symbol | Name | Formula / Description |
|---|---|---|
| r | Radius | Radius of the parent circle, must be positive |
| θ | Central angle | Angle at the center, in radians for all formulas above |
| L | Arc length | Length of the curved arc boundary: L = rθ |
| A | Sector area | Area of the pie-slice region: A = ½r²θ |
| c | Chord length | Straight-line distance between the two arc endpoints |
| S | Segment area | Area between the chord and arc: S = A − ½r²sin θ |
| P | Perimeter | Full boundary of the sector: arc + both radii = rθ + 2r |
Degree Input Equivalents
Arc length: L = π r θ° / 180
Sector area: A = π r² θ° / 360
Chord: c = 2r × sin(θ° / 2) (sin works in degrees directly)
Solving for Unknown Quantities
r from arc + θ: r = L / θ
r from area + θ: r = √(2A / θ)
θ from r + arc: θ = L / r (result in radians)
How to Use
- 1Choose solve-for mode: Pick which two values you know: r+θ (most common), Arc+θ, Area+θ, or r+Arc.
- 2Set angle unit: Select Degrees or Radians, use whichever matches your problem.
- 3Set length unit: Choose your preferred unit (cm, m, in, ft…) from the dropdown.
- 4Enter values: Type your two known values. The angle field shows ° or rad based on your unit selection.
- 5Calculate: Click Calculate or press Enter. All five outputs appear instantly with the SVG diagram.
- 6Review steps: Toggle "Show Step-by-Step" to see the full arithmetic, or copy results to clipboard.
Example Calculation
Example 1, Clock hand (r = 10 cm, θ = 60°)
θ in radians = 60 × π/180 = π/3 ≈ 1.0472 rad
Arc length = 10 × 1.0472 = 10.472 cm
Sector area = ½ × 100 × 1.0472 = 52.36 cm²
Chord = 2 × 10 × sin(30°) = 20 × 0.5 = 10.000 cm
Segment area = ½ × 100 × (1.0472 − 0.866) = 9.06 cm²
Perimeter = 10.472 + 20 = 30.472 cm
Example 2, Find radius from arc + angle
Known: arc L = 15.71 cm, angle = 90°. Solve for r.
θ = 90° = π/2 = 1.5708 rad
r = L / θ = 15.71 / 1.5708 = 10.001 cm
Sector area = ½ × 100.02 × 1.5708 = 78.55 cm²
Example 3, Semicircle (θ = 180°, r = 5 m)
Arc length = 5π ≈ 15.708 m
Sector area = πr²/2 = π×25/2 ≈ 39.270 m²
Chord = 2r = 10.000 m (diameter)
Perimeter = πr + 2r = r(π+2) ≈ 25.708 m
Understanding Circle Sector | Arc, Area & Chord
Sector geometry is foundational across many disciplines. In engineering, gear tooth profiles, cam lobes, and pipe bends all rely on accurate arc and sector calculations. In architecture, circular rooms, amphitheaters, and curved facades require sector area and chord measurements for materials estimation. In surveying, the chord formula calculates the straight-line distance between two boundary points on a curved plot boundary. In data visualization, every pie chart is built from sectors, the angle for each slice is θ = (value/total) × 2π.
What Is a Radian?
A radian is defined as the angle subtended at the center of a circle by an arc whose length equals the radius. This gives the clean formula L = rθ, no conversion factor needed. Because one full rotation is 2π radians, and 2π × r = circumference, the formula is self-consistent. Radians are the natural unit in calculus, physics, and engineering; degrees are convenient for everyday use.
Sector vs. Segment
| Region | Boundary | Area Formula |
|---|---|---|
| Sector | Two radii + arc | A = ½r²θ |
| Segment | Chord + arc | S = ½r²(θ − sin θ) |
| Triangle | Two radii + chord (straight) | T = ½r²sin θ |
| Note | Sector = Segment + Triangle | A = S + T ✓ |
Common Sector Angles
| Angle (°) | Radians | Arc / r | Area / r² | Chord / r |
|---|---|---|---|---|
| 30° | π/6 ≈ 0.5236 | 0.5236 | 0.2618 | 0.5176 |
| 45° | π/4 ≈ 0.7854 | 0.7854 | 0.3927 | 0.7654 |
| 60° | π/3 ≈ 1.0472 | 1.0472 | 0.5236 | 1.0000 |
| 90° | π/2 ≈ 1.5708 | 1.5708 | 0.7854 | 1.4142 |
| 120° | 2π/3 ≈ 2.0944 | 2.0944 | 1.0472 | 1.7321 |
| 180° | π ≈ 3.1416 | 3.1416 | 1.5708 | 2.0000 |
| 270° | 3π/2 ≈ 4.7124 | 4.7124 | 2.3562 | 1.4142 |
| 360° | 2π ≈ 6.2832 | 6.2832 | 3.1416 | 0 (full circle) |
Applications by Field
| Field | Application |
|---|---|
| Mechanical engineering | Gear tooth profile, involute curves, cam geometry |
| Civil engineering | Road curve design, bridge arch spans |
| Architecture | Circular buildings, curved facades, amphitheater seating |
| Surveying | Chord distance between boundary points on curved lots |
| Data visualization | Pie chart slice angles, θ = (value/total) × 2π |
| Irrigation | Center-pivot sprinkler coverage area calculation |
| Astronomy | Angular diameter of celestial bodies, arc-minutes/seconds |
| Clock & watch design | Hand sweep angles, hour markers at 30° intervals |
Frequently Asked Questions
What is the difference between a sector and a segment?
A sector is the "pie slice" region bounded by two radii and the arc. A segment is the smaller region between the chord and the arc (sector minus the triangle formed by the two radii). The segment area formula is S = ½r²(θ − sin θ).
How do I convert degrees to radians for the formula?
Multiply by π/180: θ_rad = θ_deg × π/180. Key conversions: 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 180°=π, 270°=3π/2, 360°=2π. The calculator handles this conversion automatically.
Can I solve for the radius if I know the arc length and angle?
Yes, select the "Arc + θ" solve mode. Enter arc length and angle, and the calculator returns radius as r = L/θ, then computes all other properties.
What is the formula for the perimeter of a sector?
Perimeter = arc length + 2 × radius = rθ + 2r = r(θ + 2). For degrees: P = r(πθ°/180 + 2). Note that the perimeter includes both straight radius edges plus the curved arc.
Why does the chord length equal the diameter for a 180° sector?
Because c = 2r × sin(θ/2) = 2r × sin(90°) = 2r × 1 = 2r, which is the diameter. A semicircle's chord is indeed a straight line across the full diameter.
What happens when the sector angle is 360°?
A full 360° sector is the complete circle. Arc length = 2πr (circumference), Sector area = πr² (full circle area), Chord length approaches 0 (the two endpoints merge), Perimeter = 2πr + 2r (but typically just cited as circumference).
How is circle sector used in pie charts?
Each slice angle θ = (category value / total) × 2π radians. For example, if a category is 25% of the total, its angle is 0.25 × 2π = π/2 = 90°. The arc length and area scale proportionally.
What is a minor vs. major sector?
A minor sector has a central angle less than 180° (smaller than a semicircle). A major sector has an angle greater than 180° (the larger piece). Together they make up the full circle area πr².