Circle Length Calculator
Enter a radius (or diameter) and a central angle to calculate the arc length, full circumference, chord length, and sector area instantly. Switch between metric and imperial units and choose degrees or radians for the angle. The calculator also reverse-solves: enter arc length and angle to find the radius, or enter circumference alone to get diameter and area.
Formula
Worked example
A circle with radius 10 cm and a central angle of 90 degrees: theta_rad = 90 x pi / 180 = pi/2 rad. Arc length L = 10 x pi/2 = 15.708 cm. Circumference C = 2 x pi x 10 = 62.832 cm. Chord length c = 2 x 10 x sin(pi/4) = 14.142 cm. Sector area A = 0.5 x 10^2 x pi/2 = 78.540 cm^2.
What is circle length?
The term "circle length" most often refers to the circumference, which is the full perimeter of a circle. When you draw a circle on paper and trace all the way around it, the total distance you cover is the circumference. The formula is C = 2 x pi x r, where r is the radius and pi is the mathematical constant 3.14159... For a partial circle, "circle length" becomes the arc length, which is the curved distance between two points on the circle's edge defined by a central angle. The arc length formula is L = r x theta, where theta is the central angle expressed in radians.
How to calculate arc length step by step
To find the arc length of a circle: (1) Identify the radius r and the central angle. (2) If the angle is in degrees, convert it to radians by multiplying by pi/180. For example, 90 degrees becomes 90 x pi/180 = pi/2 = 1.5708 rad. (3) Multiply: L = r x theta_rad. For a 10 cm radius and 90 degree angle, L = 10 x 1.5708 = 15.708 cm. The chord length, which is the straight-line distance across the arc, is c = 2r x sin(theta/2). The sector area, the pie-slice region enclosed by the two radii and the arc, is A = 0.5 x r^2 x theta_rad.
Minor arcs, major arcs and semicircles
Any chord divides a circle into two arcs. The smaller one (subtended by an angle below 180 degrees) is the minor arc, and the larger one (angle above 180 degrees) is the major arc. When the angle is exactly 180 degrees the arc is a semicircle, half the circumference. Both arcs together always equal the full circumference. If you know one arc length, subtract it from the circumference to get the other. This calculator handles all three cases: enter an angle below 180 for a minor arc, above 180 for a major arc, or exactly 360 to compute the full circumference.
Reverse-solve: finding radius or angle from arc length
Sometimes you know the arc length and need the radius or the angle. Rearranging L = r x theta: the radius is r = L / theta_rad, and the angle in radians is theta = L / r. This calculator's "Solve for" selector lets you switch modes so you can enter the known values and recover whichever quantity is unknown. You can also find the radius from the full circumference using r = C / (2 x pi), which is useful when you have measured a round object and want to confirm its radius or diameter.
Common central angles and arc fractions
| Central angle | Arc fraction | Arc length formula | Sector shape |
|---|---|---|---|
| 30 deg (pi/6 rad) | 1/12 | C / 12 | 1/12 of circle |
| 45 deg (pi/4 rad) | 1/8 | C / 8 | 1/8 of circle |
| 60 deg (pi/3 rad) | 1/6 | C / 6 | 1/6 of circle |
| 90 deg (pi/2 rad) | 1/4 | C / 4 | Quarter circle |
| 120 deg (2pi/3 rad) | 1/3 | C / 3 | Third of circle |
| 180 deg (pi rad) | 1/2 | C / 2 | Semicircle |
| 270 deg (3pi/2 rad) | 3/4 | 3C / 4 | Three-quarter circle |
| 360 deg (2pi rad) | 1 | C | Full circle |
How common central angles relate to arc length as a fraction of circumference, and the arc length formula at that angle.
Frequently asked questions
What is the difference between arc length and circumference?
Circumference is the full perimeter of a circle, the distance all the way around. Arc length is the curved distance along part of that perimeter, defined by a central angle. A 90-degree arc is one quarter of the circumference; a 180-degree arc (a semicircle) is one half. Set the central angle to 360 degrees (or 2pi radians) and the arc length equals the circumference.
How do I convert degrees to radians for the arc length formula?
Multiply the angle in degrees by pi/180 (approximately 0.017453). For example, 45 degrees = 45 x pi/180 = pi/4 = 0.7854 radians. This calculator does the conversion automatically when you choose "Degrees" for the angle unit.
What is the chord length and how is it different from arc length?
The chord is the straight line connecting the two endpoints of an arc, cutting through the interior of the circle. The arc length is the curved path along the circle's edge between those same two points. The arc is always longer than the chord (unless the angle is 0, in which case both are 0). The formula for chord length is c = 2r x sin(theta/2).
Can I find the radius if I only know the arc length and the angle?
Yes. Rearrange the arc length formula L = r x theta_rad to get r = L / theta_rad. Convert the angle to radians first. For example, an arc of 20 cm with a central angle of 60 degrees (= pi/3 rad) gives r = 20 / (pi/3) = 60/pi = 19.099 cm. Select "Radius (from arc length + angle)" in the Solve for dropdown to have the calculator do this for you.
What is sector area and how is it calculated?
A sector is the pie-slice-shaped region bounded by two radii and the arc between them. Its area is A = 0.5 x r^2 x theta_rad. For a full circle this reduces to pi x r^2, the standard circle area formula. This calculator outputs the sector area alongside arc length whenever a central angle is involved.
Does circle length depend on pi, and how precise is pi here?
Yes, all circle length formulas involve pi (the ratio of circumference to diameter, approximately 3.14159265). This calculator uses JavaScript's built-in Math.PI, which is accurate to about 15 significant digits, far more than any practical measurement requires.