Area of a Sector & Segment Calculator

Enter radius and central angle (degrees or radians) — get sector area, segment area, arc length, and step-by-step working.

Unit:
Angle:
Radius (r)
cm
Angle (θ)
°

💡 Enter radius and angle — sector & segment computed instantly.

Live Sector Diagram

Sector & Segment Formulas

Sector Area (degrees)

A = (θ/360) × πr²

Fraction of full circle's area proportional to angle.

Sector Area (radians)

A = ½r²θ

Simpler form when angle is in radians.

Arc Length (degrees)

L = (θ/360) × 2πr

In radians: L = rθ. Curved edge of the sector.

Segment Area

A = ½r²(θ − sinθ)

θ in radians. Sector minus the isosceles triangle.

Triangle Area

A = ½r²sinθ

Isosceles triangle formed by the two radii and chord.

Chord Length

chord = 2r × sin(θ/2)

Straight line connecting the two endpoints of the arc.

Sector vs Segment — What's the Difference?

Sector (Pie Slice)

A sector is bounded by two radii and the arc between them. It looks like a slice of pie. The sector includes the interior triangle plus the circular "bite" region above the chord.

Area = ½r²θ (rad)

Segment (Round Bite)

A segment is bounded by a chord and the arc. It's the region cut off when you draw a straight line across a circle. Segment = Sector − Triangle.

Area = ½r²(θ − sinθ)

The key relationship: Segment = Sector − Triangle. The triangle is the isosceles triangle formed by the two radii (each of length r) and the chord. Its area is ½r²sinθ. Subtracting it from the sector leaves the curved segment region.

Worked Example

Sector with radius 10 cm and angle 60°

Given: r = 10 cm, θ = 60°
Convert: θ = 60 × π/180 = π/3 ≈ 1.0472 rad
Arc Length: L = 10 × 1.0472 ≈ 10.47 cm
Sector Area: ½ × 100 × 1.0472 ≈ 52.36 cm²
Triangle Area: ½ × 100 × sin(60°) = 50 × 0.866 ≈ 43.30 cm²
Segment Area: 52.36 − 43.30 ≈ 9.06 cm²
Chord: 2 × 10 × sin(30°) = 20 × 0.5 = 10 cm

Degrees vs Radians

Degrees divide a full rotation into 360 equal parts — familiar from everyday use. Radians measure angle as the ratio of arc length to radius; a full circle = 2π radians ≈ 6.283. Radians are preferred in mathematics because they simplify formulas (no π/180 conversion needed).

Degrees Radians Description
30°π/6 ≈ 0.52361/12 circle
45°π/4 ≈ 0.78541/8 circle
60°π/3 ≈ 1.04721/6 circle
90°π/2 ≈ 1.5708Quarter circle
180°π ≈ 3.1416Semicircle
360°2π ≈ 6.2832Full circle

Common Sector & Segment Measurements

Radius Angle Arc Length Sector Area Segment Area
10 cm 60° 10.47 cm 52.36 cm² 9.06 cm²
10 cm 90° 15.71 cm 78.54 cm² 28.54 cm²
10 cm 120° 20.94 cm 104.72 cm² 54.72 cm²
5 cm 45° 3.93 cm 9.82 cm² 0.94 cm²
15 cm 30° 7.85 cm 58.90 cm² 2.57 cm²

Real-World Applications

🍕

Food — Pizza & Pie Portions

Pizza slices and pie chart portions are sectors. An 8-slice pizza has each slice as a 45° sector of the full circle.

🕰️

Clocks — Time Sectors

A quarter-hour is a 90° sector. Clock hands sweep sector-shaped regions as time passes — useful in mechanical watch design.

🌍

Geography — Map Projections

Conical map projections and radar sweep patterns use circular sectors to represent angular regions of the Earth's surface.

⚙️

Engineering — Fan & Gear Design

Fan blades, turbine sections, and partial gear teeth are designed as sectors. The area and arc length determine material volume and speed.

📊

Data Visualization — Pie Charts

Pie charts represent proportions as sectors. Each "slice" has a central angle proportional to its data value (e.g., 25% data = 90° sector). Understanding sector area helps verify visual data integrity.

Frequently Asked Questions

What is the difference between a sector and a segment?
A sector is the "pie slice" shape bounded by two radii and an arc — it includes the interior triangle. A segment is only the region between a chord and the arc — not including the triangle. Segment = Sector − Triangle.
What is the formula for sector area?
In degrees: A = (θ/360) × πr². In radians: A = ½r²θ. The sector is simply the fraction (θ/360) of the full circle's area. For r = 10 cm and θ = 90°: A = (90/360) × π × 100 = 25π ≈ 78.54 cm².
How do you convert between degrees and radians?
Multiply degrees by π/180 to get radians. Multiply radians by 180/π to get degrees. Key values: 360° = 2π, 180° = π, 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6.
What is arc length and how is it calculated?
Arc length is the distance along the curved edge of the sector. In radians: L = rθ. In degrees: L = (θ/360) × 2πr. For r = 10 cm and θ = 60°: L = (60/360) × 2π × 10 = (1/6) × 62.83 ≈ 10.47 cm.
How is segment area calculated step by step?
1. Convert θ to radians. 2. Find sector area = ½r²θ. 3. Find triangle area = ½r²sinθ. 4. Segment = Sector − Triangle = ½r²(θ − sinθ). For r=10, θ=60°: Sector = 52.36, Triangle = 43.30, Segment = 9.06 cm².
What happens when the angle equals 360°?
At 360°, the sector becomes a full circle with area = πr². The chord length becomes zero (both arc endpoints coincide), so the "segment" has zero area — there is no region between a chord and the arc when the chord doesn't exist.

Related Area Calculators

Circle Semicircle Square Rectangle Parallelogram Rhombus Trapezium Scalene Triangle Right-Angle Triangle Equilateral Triangle Isosceles Triangle Regular Hexagon