Guide
How to use Sector Area Calculator
Start with the working tool above. The notes below explain the inputs, method, limitations and checks that matter for this specific task.
- 1. Enter the radius in any unit — the area comes back in that unit squared.
- 2. Enter the central angle in degrees, between 0 and 360.
- 3. Read the sector area and arc length in the results.
- 4. Check the fraction shown: angle ÷ 360 is your share of the full circle.
- 5. Verify your result against the worked examples below.
How it works
A sector is a fraction of a whole circle, and that fraction is θ/360, where θ is the central angle in degrees. The full circle's area is πr², so sector area = (θ/360) × πr². The full circumference is 2πr, so arc length = (θ/360) × 2πr. Here r is the radius — half the diameter — and θ is the angle between the two radii, measured at the center. If your angle is in radians, the fraction becomes θ/(2π) and the formulas simplify to area = ½r²θ and arc = rθ, the versions used in calculus.
Worked examples
Quarter circle: r = 10, θ = 90°
Fraction = 90 ÷ 360 = 0.25. Area = 0.25 × π × 10² = 0.25 × 314.16 = 78.54 square units. Arc = 0.25 × 2π × 10 = 15.71 units — a quarter of the circumference too.
Sprinkler wedge: r = 6 m, θ = 120°
Fraction = 120 ÷ 360 = 1/3. Area = (1/3) × π × 6² = (1/3) × 113.10 = 37.70 m² watered. Arc = (1/3) × 2π × 6 = 12.57 m along the outer edge.
When this tool is useful
- Working out how much ground a rotating sprinkler covers
- Dividing pies, pizzas and cakes into equal portions
- Estimating material for wedge-shaped patios and garden beds
- Checking geometry homework involving sectors and arcs
Degrees vs radians: the same formulas in two forms
US textbooks teach the degrees form first; calculus switches to radians because the formulas get shorter. Both describe the same share of the circle — θ/360 equals θ/(2π) — so use the form matching your angle and never mix them.
| Quantity | Angle in degrees | Angle in radians |
|---|---|---|
| Sector area | (θ/360) × πr² | ½r²θ |
| Arc length | (θ/360) × 2πr | rθ |
| Full circle when | θ = 360° | θ = 2π |
Sector vs segment: the triangle makes the difference
A segment is what remains when you cut a sector with a straight chord joining the radii's endpoints. Its area is the sector's area minus that triangle: segment = (θ/360) × πr² − ½r² sin θ. For r = 10 and θ = 90°, the triangle is ½ × 10 × 10 = 50, so the segment is 78.54 − 50 = 28.54 square units. Segments appear in real problems often — the water cross-section in a partly filled pipe is one.
Common mistakes to avoid
- Using the diameter instead of the radius; doubling r quadruples the area.
- Entering a reflex angle by mistake: 270° is three-quarters of the circle, not a small slice.
- Mixing degrees and radians — 90° is π/2 radians; feeding one into the other's formula breaks the result.
- Confusing a sector with a segment — the segment is the sector minus its triangle.
- Reporting arc length in square units: arc is a length, area is in square units.
Frequently asked questions
What is the formula for the area of a sector?
For degrees, area = (θ/360) × πr², with r the radius and θ the central angle. For radians, area = ½r²θ. Both express the same fraction of the full circle, so they agree.
How do I find the arc length of a sector?
Multiply the same fraction by the circumference: arc = (θ/360) × 2πr for degrees, or arc = rθ for radians. For r = 10, θ = 90°: arc = (90/360) × 2π × 10 = 15.71.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc — the pizza slice. A segment is bounded by a chord and an arc, and its area equals the sector's area minus the triangle.
Can the angle be more than 180 degrees?
Yes. A sector between 180° and 360° is a major sector, and the same formulas apply. At 360° the sector is the whole circle; at 0° it vanishes. Most everyday slices are minor sectors, under 180°.
How do I convert degrees to radians for these formulas?
Multiply degrees by π/180: 90° = π/2 ≈ 1.5708 radians, 120° = 2π/3 ≈ 2.094. Then use area = ½r²θ and arc = rθ. Check: r = 10, θ = π/2 gives area 78.54, as before.
What if I only know the arc length and the radius?
Work backwards: θ (radians) = arc ÷ r, so an arc of 15.71 with r = 10 gives θ = 1.571 radians = 90°. Area is then ½ × r × arc = ½ × 10 × 15.71 = 78.54.
Bottom line
Every sector problem reduces to one fraction: the angle over 360. Apply it to πr² for area and to 2πr for arc length, keep degrees and radians separate, and the numbers take care of themselves.
If the question changes shape, the closest neighbours are Angle of Elevation Calculator and Tip Percentage Calculator — or browse everything under Math Calculators.