Arc Length Calculator
Calculate arc length and sector area from radius and central angle. Free online arc length calculator with visual chart, step-by-step results, and formula explanations.
About This Calculator
About Arc Length
An arc is a portion of the circumference of a circle. The arc length is the curved distance along that portion, determined by the radius of the circle and the central angle that subtends the arc. This calculator computes the arc length, sector area, and angle in radians from the radius and central angle in degrees.
Arc Length Formula
The fundamental formula for arc length is s = r x theta, where s is the arc length, r is the radius, and theta is the central angle in radians. When the angle is given in degrees, use the equivalent form s = (theta/360) x 2pir, which represents the fraction of the full circumference.
Sector Area Formula
The area of the sector (the "pizza slice" bounded by the arc and two radii) is A = (1/2) x r^2 x theta in radians, or A = (theta/360) x pir^2 in degrees. The sector area is proportional to the central angle -- a 90 deg angle gives one-quarter of the full circle's area.
Real-World Applications
Arc length calculations are used in engineering (designing curved roads, railway tracks, and pipelines), architecture (arches and domes), manufacturing (bending metal sheets and pipes), astronomy (measuring angular distances), and navigation (great-circle routes on Earth's surface).
Relationship Between Degrees and Radians
Degrees and radians are two units for measuring angles. A full circle is 360 deg or 2pi radians. To convert degrees to radians, multiply by pi/180. To convert radians to degrees, multiply by 180/pi. The calculator handles this conversion automatically -- simply enter the angle in degrees.
How to Use This Calculator
Enter the radius of the circle in any unit (e.g., meters, feet, inches) and the central angle in degrees. Click Calculate to see the arc length, sector area, and central angle in radians. The visual doughnut chart shows the arc as a portion of the full circle's circumference. All inputs are saved to the URL for easy sharing.
Edge Cases and Limitations
For a full circle (360 deg), the arc length equals the full circumference and the sector area equals the full circle area. For angles greater than 360 deg, the arc wraps around the circle multiple times. Angles between 0 deg and 360 deg produce the standard arc and sector results. The radius must be a positive number.
Frequently Asked Questions
How do you calculate arc length?
Arc length is calculated using the formula s = r x theta, where r is the radius and theta is the central angle in radians. If the angle is given in degrees, first convert it to radians by multiplying by pi/180.
What is the arc length formula in degrees?
In degrees, the arc length formula is s = (theta/360) x 2pir, where theta is the central angle in degrees and r is the radius. This gives the fraction of the circle's circumference.
How do you find the sector area from arc length?
The sector area is calculated as A = (1/2) x r^2 x theta, where r is the radius and theta is the central angle in radians. You can also use A = (theta/360) x pir^2 when working in degrees.
What is the difference between arc length and chord length?
Arc length is the curved distance along the circle between two points on its circumference. Chord length is the straight-line distance between the same two points. The arc is always longer than the chord for any angle greater than 0.
What units does the arc length calculator use?
The calculator works with any unit of length (meters, feet, inches, etc.) as long as the radius and arc length are in the same unit. The central angle is entered in degrees and is automatically converted to radians for the calculation.
How do you find the central angle if you know the arc length and radius?
If you know the arc length (s) and radius (r), the central angle in radians is theta = s/r. To convert to degrees, multiply by 180/pi.
What is a radian and why is it used in arc length formulas?
A radian is an angle measure defined as the angle whose arc length equals the radius. One full circle equals 2pi radians (≈ 6.283 rad). Radians are the natural unit for arc length because the formula s = rtheta is simpler than using degrees.
Can arc length be greater than the circumference?
Yes, if the central angle exceeds 360 deg (2pi radians), the arc length can wrap around the circle more than once. For example, a 720 deg angle produces an arc length equal to twice the circumference.