Hi there. In this math guide, I cover the arc length of a circle. Note that I do not cover the integral case.
Screenshots done on witeboard.com with the use of a Wacom tablet & pen stylus.
Topics
- Review Of Circumference Of A Circle
- Arc Length Of A Circle
- Examples
Review Of Circumference Of A Circle
The circumference of a circle is the length around the circle. You can view this as the perimeter of the circle. Given the diameter of the circle (d), the circumference formula would be C = πd.
The radius of a circle (r) is the length from one point on the outside of the circle to the middle of the circle. This radius is half the length of the diameter. Given the radius of a circle, the circumference of the circle is given by the formula C = 2πr.
Screenshot picture taken from https://www.cuemath.com/geometry/circumference-of-a-circle/.
Arc Length Of A Circle
What if do not want all of the circumference of the circle? You just want a part of it. This would be the arc length of a circle.
A full circle is related to 360 degrees or 2π in radians. Given an angle from 0 to 360 degrees or 0 to 2π in radians we can obtain the arc length of the circle. The arc length of a circle is the circumference multiplied by the ratio of the angle out of 360 degrees (or the angle divided by 2π in radians).
This symbol θ is named theta which represents the angle.
Examples
Here are a few worked through examples using the arc length formula. I use https://www.tutorialspoint.com/whiteboard.htm for the circle screenshot pictures. Witeboard.com is used for the calculations.
Example One
Determine the arc length of a circle with a radius of 5 centimetres and an angle of 30 degrees in the circular sector.
The angle θ is 30 degrees and is inputted into the arc length formula in degrees.
Example Two
Determine the arc length of a circle with a radius of 10 centimetres and an angle of pi/4 in the circular sector.
Example Three
The arc length of a circle is 10pi (10π) units. What is the angle in degrees and radians for this circular sector?
