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Arc Length Calculator
Calculate the arc length of a circle given the radius and central angle. Also computes sector area, segment area, and chord length.
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Arc Length Calculator | Circle Arc from Radius and Angle
Calculate the arc length of a circle given the radius and central angle. Also computes sector area, segment area, and chord length.
Key values: r = 10 cm · angle = 90° · arc ≈ 15.71 cm
Highway On-Ramp Curve
A highway on-ramp curves through 60° with a radius of 150 m.
Key values: r = 150 m · angle = 60° · arc ≈ 157.1 m
Arc Length Formula
An arc is a portion of a circle's circumference. Its length depends on the circle's radius and the central angle that subtends it.
When the angle is given in degrees, convert first:
The radian formula is elegant because radians are defined as the ratio of arc length to radius — so by definition.
Worked Examples
| Radius | Angle | Arc length |
|---|---|---|
| 5 cm | 90° (π/2 rad) | cm |
| 10 m | 45° (π/4 rad) | m |
| 3 ft | 120° (2π/3 rad) | ft |
| r | 360° (2π rad) | (full circumference) |
Arc Length as a Proportion
The arc length is simply a fraction of the full circumference:
This proportional relationship connects arc length, sector area, and central angle — all three share the same fraction of the whole circle.
Key relationship: If a sector has 1/4 of the central angle (90°), it also has 1/4 of the circumference (arc length) and 1/4 of the area.
Practical Applications
- Track curves: Designing curved sections of roads, railways, or running tracks requires knowing the arc length for a given turning radius.
- Belt and pulley systems: The length of belt wrapping around a pulley equals the arc length for the contact angle.
- Pizza slices: The crust length of a pizza slice is the arc length — a 12-inch pizza cut into 8 slices gives each slice inches of crust.
- Earth distances: The distance between two points on the same latitude or longitude is an arc length on a sphere with radius 6,371 km.
Frequently Asked Questions
What is arc length?
Arc length is the distance along the curved portion of a circle between two points. It is a fraction of the full circumference, determined by the central angle that subtends the arc.
How do I calculate arc length?
Use when the angle is in radians, or when the angle is in degrees. For example, a arc on a circle with radius 10 cm is cm.
What is the difference between arc length and chord length?
Arc length is the curved distance along the circle between two points, while chord length is the straight-line distance between those same two points. The arc is always longer than the chord unless the angle is zero.
How do I convert degrees to radians for the arc length formula?
Multiply degrees by . For example, radians. This conversion is necessary because the formula requires radians.
Can arc length be longer than the circumference?
No. The maximum arc length equals the full circumference (when the central angle is or radians). An arc is always a portion of the circumference, so it ranges from 0 up to .
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