Sector Area Calculator

Find the area, arc length, chord, and perimeter of a circular sector from radius and angle.

🏧 What is a Sector of a Circle?

A sector is the region of a circle bounded by two radii and the arc between them — it looks like a “pie slice.” The two straight edges are radii of the circle (each of length r), and the curved edge is the arc. The central angle (θ) is the angle between the two radii at the center of the circle. A full circle is a sector with θ = 360°; a semicircle has θ = 180°; a quarter circle has θ = 90°.

The area of a sector is a fraction of the full circle area: Area = (θ/360) × πr² when θ is in degrees. In the cleaner radian form: Area = ½r²θ. The arc length is the corresponding fraction of the circumference: arc = (θ/360) × 2πr = rθ (radians). Both formulas reflect the same idea: the sector is proportional to its central angle.

Three additional measurements are useful: the chord (the straight line joining the arc’s endpoints, with length 2r × sin(θ/2)); the sector perimeter (arc + 2r); and the segment area (sector area minus the isosceles triangle formed by the two radii and chord). This calculator provides sector area, arc, chord, and perimeter — giving you all the information needed for typical geometry and engineering problems.

Sectors appear everywhere: pie charts (each slice is a sector whose angle is proportional to the data percentage); sprinkler heads (the irrigated area is a sector defined by throw radius and rotation angle); clock hands (the area swept in a time period); fan coverage; gear tooth profiles; and segment mirrors in optical systems. Understanding sector geometry is essential for any problem involving circular coverage areas or proportional circular regions.

📐 Formulas

Area = ½r²θ (or (θ/360) × πr² in degrees)

r = radius of the circle · θ = central angle in radians

Arc length = rθ · Chord = 2r × sin(θ/2) · Perimeter = rθ + 2r

Example (r=5, θ=90°=π/2 rad): Area = ½ × 25 × π/2 = 25π/4 ≈ 19.635 · Arc = 5π/2 ≈ 7.854

📖 How to Use This Calculator

Steps

Select Degrees or Radians for your angle input.

Enter the radius (the distance from the center to the arc, in any unit) and the central angle (the angle between the two radii at the circle center).

Click Calculate to see sector area, arc length, chord length, and total perimeter.

💡 Example Calculations

Example 1 — Quarter Circle (90°)

Radius = 5 cm, angle = 90°

Area = (90/360) × π × 25 = 25π/4 ≈ 19.635 cm²

Arc = (90/360) × 2π × 5 = 5π/2 ≈ 7.854 cm. Chord = 2 × 5 × sin(45°) = 10/√2 ≈ 7.071 cm.

Area ≈ 19.635 cm². Perimeter = 7.854 + 10 ≈ 17.854 cm.

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Example 2 — 60° Sector (Equilateral Triangle Connection)

Radius = 10 cm, angle = 60°

Area = (60/360) × π × 100 = 100π/6 ≈ 52.36 cm²

Chord = 2 × 10 × sin(30°) = 20 × 0.5 = 10 cm = radius! Forms an equilateral triangle.

Area ≈ 52.36 cm². For 60°: chord = radius — the two radii and chord form an equilateral triangle.

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Example 3 — Sprinkler Coverage (120° arc)

Sprinkler throw radius = 4 m, rotation = 120°

Area = (120/360) × π × 16 = 16π/3 ≈ 16.76 m²

Arc = (120/360) × 2π × 4 = 8π/3 ≈ 8.378 m

Coverage area ≈ 16.76 m². To cover a 50 m² zone, you would need ≈ 3 such sprinklers (non-overlapping).

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Example 4 — Radian Mode (θ = π/3 rad)

Radius = 8 cm, angle = π/3 ≈ 1.0472 radians (= 60°)

Area = ½ × 64 × π/3 = 32π/3 ≈ 33.51 cm²

Arc = 8 × π/3 = 8π/3 ≈ 8.378 cm. Chord = 2 × 8 × sin(π/6) = 16 × 0.5 = 8 cm

Area ≈ 33.51 cm². Note: arc (8.378 cm) > chord (8 cm) — arc always ≥ chord.

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Frequently Asked Questions

What is the formula for the area of a sector?

Area of sector = (θ/360) × π × r² when θ is in degrees, or ½ × r² × θ when θ is in radians. Both formulas give the same result. Example: radius = 5 cm, angle = 90°: Area = (90/360) × π × 25 = (1/4) × π × 25 = 25π/4 ≈ 19.635 cm².

How do you calculate arc length?

Arc length = (θ/360) × 2πr in degrees, or simply = r × θ in radians. Example: radius = 8 m, angle = 60°: arc = (60/360) × 2π × 8 = (1/6) × 16π = 8π/3 ≈ 8.378 m. The arc is proportional to the angle — twice the angle gives twice the arc.

What is the difference between an arc, sector, and segment?

Arc: the curved portion of the circle's circumference between two points. Sector: the 'pie slice' bounded by two radii and an arc (includes the interior). Segment: the region bounded by a chord and the arc it subtends (does NOT include the center). Segment area = sector area − triangle area.

How do I convert degrees to radians for the sector formula?

Radians = degrees × π/180. Common conversions: 30° = π/6 ≈ 0.5236 rad; 45° = π/4 ≈ 0.7854 rad; 60° = π/3 ≈ 1.0472 rad; 90° = π/2 ≈ 1.5708 rad; 180° = π ≈ 3.1416 rad; 360° = 2π ≈ 6.2832 rad. This calculator handles the conversion automatically — just enter the angle in whichever unit you prefer.

What is the chord length of a sector?

Chord length = 2r × sin(θ/2), where θ is the central angle in radians. The chord is the straight line connecting the two endpoints of the arc. Example: r = 10 cm, θ = 60° = π/3 rad: chord = 2 × 10 × sin(30°) = 20 × 0.5 = 10 cm. (For a 60° sector with radius r, the chord equals r — it forms an equilateral triangle with the two radii.)

What is the perimeter of a sector?

Perimeter = arc length + 2 × radius. The two straight sides are the radii (each of length r), and the curved side is the arc. Example: r = 5, θ = 90°: arc = (90/360) × 2π × 5 = 2.5π ≈ 7.854. Perimeter = 7.854 + 10 ≈ 17.854 units.

What is the area of a semicircle sector?

A semicircle is a sector with θ = 180° (π radians). Area = ½πr². For radius 4 cm: area = ½ × π × 16 = 8π ≈ 25.13 cm². Arc length = πr = 4π ≈ 12.57 cm. Perimeter = πr + 2r = r(π + 2) ≈ 4 × 5.142 ≈ 20.57 cm.

How is sector area used in real life?

Common applications: (1) Pie chart segments — each sector's area is proportional to its percentage. (2) Sprinkler irrigation coverage — a rotating sprinkler covers a sector of radius equal to the water throw distance. (3) Clock hands — the region swept by a clock hand in a given time is a sector. (4) Windscreen wiper coverage area. (5) Fan blade swept area. (6) Cam and gear design in mechanical engineering.

What is the area of a sector with radius 6 and angle 120°?

Area = (120/360) × π × 6² = (1/3) × π × 36 = 12π ≈ 37.699 square units. Arc length = (120/360) × 2π × 6 = (1/3) × 12π = 4π ≈ 12.566. Chord = 2 × 6 × sin(60°) = 12 × (√3/2) = 6√3 ≈ 10.392. Perimeter = 4π + 12 ≈ 24.566.

How do you find the radius of a sector given its area and angle?

From A = (θ/360)πr² in degrees: r = √(360A / (πθ)). From A = ½r²θ in radians: r = √(2A/θ). Example: area = 50 cm², angle = 72°: r = √(360 × 50 / (π × 72)) = √(18000/226.19) = √79.58 ≈ 8.92 cm.

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📌 Quick Tips

💡A sector is a 'pie slice' of a circle. A full circle is a sector with angle 360° (or 2π radians). A semicircle is a sector with angle 180° (or π radians).

💡The sector area formula in radians is simpler: A = ½r²θ. In degrees: A = (θ/360) × πr². Both give the same result.

💡Arc length = rθ (in radians) = (θ/360) × 2πr (in degrees). The arc is the curved boundary of the sector.

💡The chord is the straight line connecting the two endpoints of the arc. Chord length = 2r × sin(θ/2) where θ is in radians.

💡The perimeter of a sector = arc length + 2 × radius (the two straight radii plus the curved arc).