Arc Length Calculator
Calculate the exact mathematical arc length of a circle segment. Enter any two known variables (Radius, Angle, or Length) to instantly solve for the third, complete with step-by-step mathematical proofs.
How to Calculate Arc Length
An arc length is simply a fraction of the circumference of a circle. Because of this, the formula you use depends entirely on how you are measuring the fraction—whether you are using standard degrees (°), or mathematical radians (rad).
Formula for Radians
Where s is Arc Length, r is Radius, and θ is the angle measured strictly in radians.
What if my angle is in Degrees?
When dealing with degrees, you have to convert the formula. Since a full circle is 360 degrees, and the full perimeter of a circle is 2πr, you multiply the perimeter by the fraction of the circle you are cutting out (θ/360).
Formula for Degrees
Which algebraically simplifies down to: s = πrθ / 180.
The Relationship Between Radius and Angle
It is vital to understand that the arc length scales linearly with the radius. If you have a pizza slice with a 45-degree angle crust, and you order an extra-large pizza with the exact same 45-degree angle, the actual crust (arc length) grows significantly larger simply because the radius increased. Angle dictates the shape, but radius dictates the actual measured distance.
Frequently Asked Questions
Studying for Geometry or Calculus Exams?
Share this step-by-step calculator with your study group or classmates to instantly verify homework answers.
Suggested hashtags: #Geometry #Calculus #MathHelp #StudyTools #thecalcs