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Area of Sector
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Sources & Methodology
Formulas verified against Khan Academy geometry curriculum and standard mathematics references for circle sector calculations.
Khan Academy — Area of Sectors
Standard sector area formulas in degrees and radians; arc length derivation and worked examples
Math Is Fun — Circle Sector and Segment
Sector area, arc length, perimeter formulas and the difference between sector and segment explained with visuals
Methodology: Degrees mode: Area = (θ/360) × π × r²; Arc length = (θ/360) × 2πr. Radians mode: Area = ½ × r² × θ; Arc length = r × θ. Perimeter = Arc length + 2r. Fraction of full circle = θ/360 (degrees) or θ/(2π) (radians). All inputs are validated; angles are converted internally using Math.PI. Results rounded to 4 decimal places.
⏱ Last reviewed: March 2026
How to Calculate the Area of a Sector
A sector is a “pie slice” portion of a circle, bounded by two radii and an arc. The central angle determines what fraction of the full circle the sector occupies, which directly determines its area. There are two equivalent formulas depending on whether your angle is in degrees or radians.
Area of Sector Formula — Degrees
Area = (θ ÷ 360) × π × r²
Where θ is the central angle in degrees and r is the radius.
Example: r = 5, θ = 90°
Area = (90/360) × π × 25 = 0.25 × 78.540 = 19.635 sq units
Example: r = 5, θ = 90°
Area = (90/360) × π × 25 = 0.25 × 78.540 = 19.635 sq units
Area of Sector Formula — Radians
Area = ½ × r² × θ
Where θ is the central angle in radians and r is the radius.
Example: r = 5, θ = π/2 (= 1.5708 rad = 90°)
Area = ½ × 25 × 1.5708 = 19.635 sq units
Example: r = 5, θ = π/2 (= 1.5708 rad = 90°)
Area = ½ × 25 × 1.5708 = 19.635 sq units
Arc Length & Perimeter of a Sector
Arc length (degrees) = (θ ÷ 360) × 2πr
Arc length (radians) = r × θ
The perimeter of the sector = arc length + 2r (arc + two straight radii).
Example: r = 5, θ = 90°
Arc = (90/360) × 2π×5 = 0.25 × 31.416 = 7.854
Perimeter = 7.854 + 10 = 17.854 units
Example: r = 5, θ = 90°
Arc = (90/360) × 2π×5 = 0.25 × 31.416 = 7.854
Perimeter = 7.854 + 10 = 17.854 units
Common Sector Angles — Reference Table
| Angle | Radians | Fraction | Area (r=5) | Arc (r=5) |
|---|---|---|---|---|
| 30° | π/6 | 1/12 | 6.545 | 2.618 |
| 45° | π/4 | 1/8 | 9.817 | 3.927 |
| 60° | π/3 | 1/6 | 13.090 | 5.236 |
| 90° | π/2 | 1/4 | 19.635 | 7.854 |
| 120° | 2π/3 | 1/3 | 26.180 | 10.472 |
| 180° | π | 1/2 | 39.270 | 15.708 |
| 270° | 3π/2 | 3/4 | 58.905 | 23.562 |
| 360° | 2π | 1 | 78.540 | 31.416 |
Sector vs Segment — Key Difference
A sector is the entire pie-slice shape — bounded by two radii and an arc, with the pointed center vertex included. A segment is just the region between a chord and its arc, like a pie slice with the pointy tip removed. To find segment area: Segment = Sector area − Triangle area (the triangle formed by the two radii and chord).
Minor Sector vs Major Sector
When the central angle is less than 180°, the sector is a minor sector (smaller than a semicircle). When it exceeds 180°, it is a major sector. The two always add up to the full circle: Minor area + Major area = πr². A 90° sector is a quarter-circle, a 180° sector is a semicircle, and a 360° sector is the complete circle.
💡 Why the radian formula looks simpler: Area = ½r²θ has no π because radians are defined as the ratio of arc length to radius, so the π factor is already built into the angle. The degrees formula needs to divide by 360 and multiply by π to achieve the same result. Both are mathematically identical.
Frequently Asked Questions
For degrees: Area = (θ/360) × π × r². For radians: Area = ½ × r² × θ. Both formulas compute the same result — they just use different angle units. The fraction θ/360 (degrees) or θ/2π (radians) tells you what portion of the full circle the sector occupies, and you multiply that by the full circle area πr².
Divide the central angle by 360 to get the fraction of the full circle, then multiply by πr². For example, r = 5 cm, θ = 90°: fraction = 90/360 = 0.25; full circle area = π×25 = 78.54; sector area = 0.25 × 78.54 = 19.63 cm².
Use Area = ½ × r² × θ. For radius = 5 and θ = π/2 radians: Area = ½ × 25 × 1.5708 = 19.635 square units. This formula is often preferred in calculus and advanced mathematics because it is compact and has no conversion factor.
Arc length = (θ/360) × 2πr for degrees, or arc length = r × θ for radians. It is the curved boundary of the sector. The full perimeter of a sector is arc length + 2r, since you also have two straight radii forming the straight sides.
A sector is the complete pie-slice including the center point, bounded by two radii and an arc. A segment is the region between a chord and an arc — like a sector with the triangular part near the center removed. Segment area = Sector area − (area of the triangle formed by the two radii and chord).
Multiply degrees by π/180. Common conversions: 30° = π/6 ≈ 0.5236, 45° = π/4 ≈ 0.7854, 60° = π/3 ≈ 1.0472, 90° = π/2 ≈ 1.5708, 180° = π ≈ 3.1416, 360° = 2π ≈ 6.2832.
A minor sector has central angle less than 180° (smaller than a semicircle). A major sector has central angle greater than 180° (larger than a semicircle). Together they always sum to the full circle area πr². A 90° sector is a quarter-circle; 180° is a semicircle.
A sector is a proportional fraction of a full circle. The fraction = θ/360 in degrees or θ/2π in radians. A 90° sector is exactly ¼ of the circle, a 120° sector is ⅓, a 180° sector is ½, and a 360° sector is the entire circle. This is why both area and arc length formulas start by computing that fraction.
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