Arc Length Calculator
Enter the radius and one other measurement of the arc: its angle in degrees or radians, its length or its sector area. The rest is worked out for you.
Worked examples
A quarter circle
Radius 10, angle 90 degrees
Arc 15.708, sector 78.540, chord 14.142
A quarter of the circumference of 62.832.
Finding the angle
Radius 5, arc length 7.854
90 degrees
7.854 ÷ 5 = 1.5708 radians.
About Arc Length Calculator
An arc is a piece of the edge of a circle, and a sector is the slice between the arc and the two radii that end it. If the angle at the centre is θ in radians, the arc length is the radius times θ and the sector area is half the radius squared times θ. In degrees, the arc is the fraction θ over 360 of the whole circumference, and the sector is the same fraction of the whole area. These come up in pie charts, curved paths, wheels and gears.
The chord is the straight line joining the two ends of the arc, with a length of 2r sin(θ/2). The segment is the area between the chord and the arc, which equals the sector minus the triangle, half r² times (θ - sin θ). Enter the radius and any one of the angle in degrees, the angle in radians, the arc length or the sector area, and the page finds the rest, including the perimeter of the sector and the height of the arc above its chord. A quarter circle of radius 10 has an arc of 15.708 and a chord of 14.142.
How to use Arc Length Calculator
- Enter the radius of the circle.
- Choose what else you know: the angle in degrees or radians, the arc length or the sector area.
- Enter that value. For radians you can type things like pi/2.
- Read the arc, sector, chord and segment, and the working.
Common uses
- Curved garden paths and edging lengths
- Pie chart and wedge areas
- Cutting arched windows and doorways
- Circle sector homework and exam questions
Frequently asked questions
What is the formula for arc length?
In radians it is radius × angle. In degrees it is the angle divided by 360, times 2 × pi × radius.
What is the difference between a sector and a segment?
A sector is a slice from the centre, like a piece of pie. A segment is the part between a chord and its arc, like the cap left when a chord cuts off part of the circle.
How do I convert degrees to radians?
Multiply by pi and divide by 180. So 90 degrees is pi/2 radians, which is about 1.5708.
What is the biggest angle I can enter?
A full turn, 360 degrees or 2 × pi radians. Beyond that the arc would wrap round the circle more than once.
What is the height of the arc above the chord?
It is also called the sagitta, and it is the distance from the middle of the chord to the middle of the arc. It is r × (1 - cos(θ/2)).