Arc Length Calculator
Find the arc length of a circle from the radius and central angle, plus the sector area, chord length and segment area, with steps.
Results
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Steps
| Step | Working |
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Quick Answer
Arc length = radius × angle in radians, or s = rθ. In degrees, s = 2πr × (angle ÷ 360). A circle with radius 10 and a 60° angle has an arc length of 10.4720 and a sector area of 52.3599.
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Table of Contents
- Arc Length Calculator
- Quick Answer
- Related Calculators
- How This Arc Length Calculator Works
- How to Use the Arc Length Calculator
- Core Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- Arc Length vs Chord Length
- Arc Length for a Radius of 1
- Tips
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Arc Length Calculator Works
Enter the radius and the central angle, and choose degrees or radians. The calculator converts the angle to radians and finds the arc length, the area of the sector (the pie slice), the straight-line chord between the arc’s ends, and the segment area between the chord and the arc.
Quick explanation: an arc is a part of a circle’s edge. Its length grows with both the radius and the angle.
- Arc length: the curved distance along the circle.
- Chord: the straight distance between the two ends.
- Sector: the slice between two radii and the arc.
How to Use the Arc Length Calculator
- Enter the radius of the circle.
- Enter the central angle.
- Choose degrees or radians.
- Press Calculate.
- Read the arc length, sector area, chord and the steps.
Core Formulas
- Arc length: s = r × θ (θ in radians)
- In degrees: s = 2πr × (angle ÷ 360)
- Sector area: A = r²θ ÷ 2
- Chord length: c = 2r × sin(θ ÷ 2)
- Segment area: r²(θ − sin θ) ÷ 2
Key Terms
- Arc: a part of the circumference of a circle.
- Central angle: the angle at the center of the circle between two radii.
- Radian: the angle where the arc length equals the radius. 2π radians = 360°.
- Sector: the region between two radii and an arc.
- Chord: a straight line joining two points on a circle.
Worked Example
Find the arc length for a radius of 5 and an angle of 120°.
- θ = 120 × π ÷ 180 = 2.0944 radians
- s = 5 × 2.0944 = 10.4720
- Sector area = 25 × 2.0944 ÷ 2 = 26.1799, and the chord is 8.6603
Real-Life Examples
- A curved path: a garden path along a 30 ft radius curve through 45° is 23.56 ft long.
- A pizza slice: a 7 inch radius pizza cut into 8 slices has a crust edge of 5.50 in per slice and an area of 19.24 sq in.
- A clock: the tip of a 6 cm minute hand moves 9.42 cm in 15 minutes (90°).
- A curved wall: a 90° curve with a 12 ft radius needs 18.85 ft of trim.
Arc Length vs Chord Length
- Arc length: the distance along the curve.
- Chord length: the straight-line distance between the ends, which is always shorter.
- Small angles: the two are almost equal when the angle is small.
Arc Length for a Radius of 1
Multiply the arc by your radius, and the area by your radius squared.
| Angle | Radians | Arc length | Sector area |
|---|---|---|---|
| 30° | 0.5236 | 0.5236 | 0.2618 |
| 45° | 0.7854 | 0.7854 | 0.3927 |
| 60° | 1.0472 | 1.0472 | 0.5236 |
| 90° | 1.5708 | 1.5708 | 0.7854 |
| 120° | 2.0944 | 2.0944 | 1.0472 |
| 180° | 3.1416 | 3.1416 | 1.5708 |
| 270° | 4.7124 | 4.7124 | 2.3562 |
| 360° | 6.2832 | 6.2832 | 3.1416 |
Tips
- Use radians: the formula s = rθ only works with the angle in radians.
- Full circle: 360° = 2π radians, so the arc is the whole circumference.
- Semicircle: 180° gives half the circumference, πr.
- Units: the arc is in the same unit as the radius.
Common Mistakes to Avoid
- Using degrees in s = rθ. Convert to radians first, or use the degree formula.
- Using the diameter as the radius. The radius is half the diameter.
- Mixing up arc and chord. The arc follows the curve; the chord is straight.
- Forgetting units. The area is in square units.
- Rounding π too early. Keep full precision until the end.
Frequently Asked Questions
Multiply the radius by the angle in radians: s = rθ.
s = 2πr × (angle ÷ 360).
Multiply by π and divide by 180.
A = r²θ ÷ 2 with θ in radians, or πr² × (angle ÷ 360) in degrees.
Divide the arc length by the radius to get the angle in radians: θ = s ÷ r.
Divide the arc length by the angle in radians: r = s ÷ θ.
c = 2r × sin(θ ÷ 2).
Yes. 180° is π radians, so s = πr.
The region between a chord and the arc above it.
Yes for arc length, which then wraps around more than once. Chord and segment use the angle within one turn.
It is in the same unit as the radius, such as inches or meters.
The circumference, 2πr.
Measure the chord across the curve and the height of the arc at its middle (the sagitta). Then radius = h ÷ 2 + c² ÷ (8h), where c is the chord and h is the height.
A radian is defined so that an angle of 1 radian makes an arc equal to the radius. That is why the arc length is simply the radius times the angle.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Arc length | Arc length formulas. | https://en.wikipedia.org/wiki/Arc_length |
| Wikipedia, Circular sector | Sector area and chord. | https://en.wikipedia.org/wiki/Circular_sector |