Arc Length and Sector Area Calculator (Free Step-by-Step)
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An arc length and sector area calculator uses two formulas with the angle θ in radians: arc length s = rθ and sector area A = ½r²θ. For a radius of 6 and an angle of 60° (π/3), the arc length is 6 × π/3 = 2π ≈ 6.28 and the sector area is ½ × 36 × π/3 = 6π ≈ 18.85 square units.
A sector with radius 6 and central angle 60° has arc length 2π ≈ 6.28 and area 6π ≈ 18.85 square units.
Chord, segment and the rest of the circle
Show steps
- Convert the angle to radians: θ = 60° × π/180 = π/3 ≈ 1.05 rad. The formulas s = rθ and A = ½r²θ need radians.
- Arc length: s = rθ = 6 × π/3 = 2π ≈ 6.28.
- Sector area: A = ½r²θ = ½ × 36 × π/3 = 6π ≈ 18.85 square units.
- Same answer with degrees: s = (60/360) × 2πr and A = (60/360) × πr². The sector is 1/6 of the whole circle.
- Chord: c = 2r × sin(θ/2) = 2 × 6 × sin(30°) = 6.
- Segment area (between the chord and the arc) = sector area − triangle area = ½r²θ − ½r² sin θ = 6π − 9√3 ≈ 3.26 square units.
- Sector perimeter = r + r + s = 2 × 6 + 2π = 12 + 2π ≈ 18.28.
Keep π in the answer for an exact value. Round only at the end.
How to use the arc length and sector area calculator
- Pick the two values you know: radius and angle, radius and arc, radius and area, angle and arc, angle and area, or arc and area.
- Type them in. Choose degrees or radians for the angle. Radians and lengths can be multiples of π, such as π/3 or 3π.
- Read the arc length, sector area, radius and angle as exact values with π and as decimals. Below them you get the chord, segment area, sector perimeter, segment height and the major arc and sector.
- Check the diagram and open Show steps to see each formula with your numbers.
Arc length and sector area formulas
A sector is a “pizza slice” of a circle: two radii and the arc between them. The arc length s is the curved edge, the sector area A is the space inside, and θ is the central angle.
| Measurement | θ in radians | θ in degrees |
|---|---|---|
| Arc length s | s = rθ | s = (θ/360) × 2πr |
| Sector area A | A = ½r²θ | A = (θ/360) × πr² |
| Area from the arc | A = ½rs | A = ½rs |
| Chord c | c = 2r sin(θ/2) | c = 2r sin(θ/2) |
| Segment area | ½r²(θ − sin θ) | (θ/360)πr² − ½r² sin θ |
| Sector perimeter | 2r + s | 2r + s |
The degree forms say the same thing: a sector is the fraction θ/360 of the whole circle, so it gets that fraction of the circumference and of the area. To switch, multiply degrees by π/180 to get radians.
Finding the radius or the angle
Any two of r, θ, s and A fix the sector. Rearrange the two main formulas (θ in radians):
- From r and s: θ = s ÷ r, then A = ½rs.
- From r and A: θ = 2A ÷ r², then s = 2A ÷ r.
- From θ and s: r = s ÷ θ.
- From θ and A: r = √(2A ÷ θ).
- From s and A: r = 2A ÷ s, then θ = s ÷ r = s² ÷ (2A).
The angle can never be more than 2π (360°). If your numbers give a bigger angle, the arc is longer than the whole circle, so one of the values is wrong.
Worked examples
Example 1: radius 6, angle π/3
s = rθ = 6 × π/3 = 2π ≈ 6.28. A = ½r²θ = ½ × 36 × π/3 = 6π ≈ 18.85 square units. The angle π/3 is 60°, one sixth of the circle, and the circle’s area is 36π, so the check 36π ÷ 6 = 6π agrees.
Example 2: radius 4 cm, angle 90°
A quarter circle. Arc length = (90/360) × 2π × 4 = 2π ≈ 6.28 cm. Sector area = (90/360) × π × 16 = 4π ≈ 12.57 cm². The chord is 4√2 ≈ 5.66 cm.
Example 3: radius 9, angle 120°
The sector is 120/360 = 1/3 of the circle. Arc length = (1/3) × 18π = 6π ≈ 18.85. Sector area = (1/3) × 81π = 27π ≈ 84.82 square units.
Example 4: radius 5, sector area 25
θ = 2A ÷ r² = 50 ÷ 25 = 2 radians (about 114.59°). Then s = rθ = 10. The shortcut s = 2A ÷ r = 50 ÷ 5 = 10 gives the same arc without finding the angle first.
Example 5: chord and segment
For r = 12 and θ = 60°: the chord is 2 × 12 × sin 30° = 12. The sector area is ½ × 144 × π/3 = 24π. The triangle under the chord has area ½ × 144 × sin 60° = 36√3. The segment between the chord and the arc is 24π − 36√3 ≈ 13.04 square units.
Common angles on a unit circle
| Angle | Radians | Share of circle | Arc (r = 1) | Sector area (r = 1) |
|---|---|---|---|---|
| 30° | π/6 | 1/12 | π/6 ≈ 0.52 | π/12 ≈ 0.26 |
| 45° | π/4 | 1/8 | π/4 ≈ 0.79 | π/8 ≈ 0.39 |
| 60° | π/3 | 1/6 | π/3 ≈ 1.05 | π/6 ≈ 0.52 |
| 90° | π/2 | 1/4 | π/2 ≈ 1.57 | π/4 ≈ 0.79 |
| 120° | 2π/3 | 1/3 | 2π/3 ≈ 2.09 | π/3 ≈ 1.05 |
| 180° | π | 1/2 | π ≈ 3.14 | π/2 ≈ 1.57 |
| 270° | 3π/2 | 3/4 | 3π/2 ≈ 4.71 | 3π/4 ≈ 2.36 |
| 360° | 2π | 1 | 2π ≈ 6.28 | π ≈ 3.14 |
For another radius r, multiply the arc by r and the sector area by r².
Common mistakes
- Using degrees in s = rθ. With r = 6 and 60°, s = 6 × 60 = 360 is far too big. Convert 60° to π/3 first.
- Mixing up arc and chord. The arc is curved and always a little longer than the straight chord between the same two points.
- Forgetting the ½ in A = ½r²θ. Without it, the area of a semicircle would come out as the whole circle.
- Squaring the wrong thing. Only r is squared in the area formula, not θ.
- Sector vs segment. The sector includes the triangle down to the center. The segment is only the piece between the chord and the arc.
Where arcs and sectors show up
A 14-inch pizza cut into 8 slices has r = 7 and θ = 45°, so each slice has 7π/4 ≈ 5.50 inches of crust and 49π/8 ≈ 19.24 in² of area. The tip of a 10 cm minute hand moves 120° in 20 minutes and travels 20π/3 ≈ 20.94 cm. Engineers use arc length for curved roads, rails and bent pipes, and sector area for sprinkler coverage, fan-shaped rooms and pie charts.
Arc Length and Sector Area practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Convert first: 30° = π/6. Then s = 12 × π/6 = 2π. Multiplying 12 by 30 uses degrees in a radian formula.
120° is 1/3 of the circle, and the circle's area is 9π, so the sector is 9π ÷ 3 = 3π ≈ 9.42.
c = 2r sin(θ/2) = 2 × 8 × sin 30° = 8. The arc, 8π/3 ≈ 8.38, is a little longer than the chord.
Sector = ½ × 16 × π/2 = 4π. Triangle = ½ × 16 × sin 90° = 8. Segment = 4π − 8 ≈ 4.57.
The perimeter is two radii plus the arc: 5 + 5 + 2π = 10 + 2π ≈ 16.28.
Quiz complete
Frequently asked questions
How do you calculate arc length and sector area?
Write the angle in radians, then use s = rθ and A = ½r²θ. For r = 6 and θ = π/3, the arc is 2π ≈ 6.28 and the area is 6π ≈ 18.85. With degrees, use (θ/360) × 2πr and (θ/360) × πr².
What is the difference between arc length and sector area?
Arc length is a distance along the curved edge, measured in units such as cm. Sector area is the space inside the slice, measured in square units such as cm². They are linked by A = ½rs.
What is the arc length and sector area formula in degrees?
Arc length = (θ/360) × 2πr and sector area = (θ/360) × πr². For a 120° sector of radius 9, that gives 6π ≈ 18.85 and 27π ≈ 84.82.
How do you find the arc length of a sector from its area?
Use A = ½rs, so s = 2A ÷ r. A sector with area 25 and radius 5 has an arc of 50 ÷ 5 = 10. If you know the angle instead of the radius, find r = √(2A ÷ θ) first.
How do you find arc length without the angle?
You need two other measurements. With the radius and the sector area, s = 2A ÷ r. With the radius and the chord c, find the angle from θ = 2 arcsin(c ÷ 2r), then s = rθ.
Does the angle have to be in radians?
For s = rθ and A = ½r²θ, yes. The degree versions include the conversion inside the θ/360 factor. The calculator accepts either and shows both.
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