Arc Length Calculator

The arc length is the curved distance along a portion of the circumference of a circle. It is the distance traveled by going along the edge of an arc, not the straight-line distance across the arc.

This calculator will calculate the arc length given your known values for any two variables in the formula. Enter any two values into the text boxes to find the third value. The chord, diameter and side of the sector are also displayed.

What is an arc length?

An arc is a portion of the circumference of a circle. Its length is determined by two factors: the size of the circle, which is set by the radius, and the angle of opening of the sector, which is determined by the central angle.

As the radius increases, so does the arc length, and as the angle of opening increases, the arc also becomes longer. When the angle reaches a full rotation and measures 360 degrees, then the arc is equal to the entire circumference of the circle.

Formula for arc length:

If the angle is given in radians then the arc length is simply the product of the radius and the angle.

If you want to use the degree method, then you can do your calculations during the conversion. A half circle is equivalent to π radians, or 180 degrees. So multiply by π and divide by 180.

Consider a circle with a radius of 5 meters and a central angle of 60 degrees. To convert this to radians we multiply by pi and divide by 180, which gives us approximately 1.047.

Multiplying that by the radius gives us an arc length of about 5.24 meters.

The area of a sector, or the portion of a circle enclosed by an arc and two radii, is equal to the product of the length of the arc and the radius divided by 2. In our example, this would be approximately 13.1 square meters. Doubling the radius will double the length of the arc, and doubling the angle will also double the length of the arc.

How to use this calculator:

Enter any two values and select the corresponding units. For the angle you can choose degrees or radians, for length meters or yard-pounds. When you enter the second value the remaining two main fields will be automatically filled in, and the arc length will be shown as the main result. The order of entry doesn't matter. You can use combinations of radius and angle, angle and arc length, or area of sector and radius. If you want to start with a different pair of numbers just clear one field and enter a new value.

If you check the larger arc option, you can also see the larger arc. This is the part of the circle that remains after cutting off the sector, along with its corresponding angle. By unfolding the compare section, you can observe how the arc length and chord change as the angle becomes larger.

How to Calculate Arc Length Without the Radius.

The radius is not always required. If the area of the sector and central angle are known, then the arc length can be calculated by taking twice the area and multiplying it with the angle in radians, then taking the square root. Alternatively, you can combine the area of the sector and the radius. In this case, the arc length is calculated by dividing twice the area by the radius. This calculator automatically chooses the correct method based on the values entered.

Various measurement values of the bow

Once the radius and angle are known all other measurements can be derived. The following table shows the individual formulas used by the calculator. Angles are always given in radians.

Examples of calculations

The following lines show the arc lengths for various angles when the radius is 10 units. The results are rounded to two decimal places. As the arc length is simply the product of the radius and the angle (in radians), it increases with the angle.

Note that for small angles the chord is only slightly shorter than the arc length but as the sector becomes larger this difference becomes more and more. At 180 degrees the chord already corresponds to a diameter, so a straight line of length 20 while the arc length extends to half of the entire circumference.

Applications of arc length

Road and rail planners use arc length when planning curves to ensure that the tracks transition into the next segment with the correct radius. Machine designers also use arc length when milling gears, shaping cams or determining dimensions of belts that go around a pulley. Physicists use arc length by multiplying the radius by the angle in radians swept out to calculate the linear distance traveled by a point on a spinning disk. Surveyors and navigators measure large circular arcs that cross the earth since a straight line does not follow the surface of the earth. Sheet metal and woodworkers need to know the arc length when laid out before cutting materials that will be bent into an arc.

Small arch and large arch

Any two points on a circle divide the circle into two arcs. The shorter arc is called the minor arc and has an angle of less than 180 degrees. The longer, more circuitous arc is called the major arc, whose angle is the supplement to the first arc's angle, calculated by subtracting the first arc's angle from 360 degrees. Since the two arcs together make up the entire circumference, the major arc is always equal to the circumference minus the minor arc. The "Major Arc" option allows both to be shown at once.

Tips for accurate results

Make sure the units of angle and length are consistent. If the angle is given in degrees and the radius is in meters, then the unit for arc length will be meters. If the calculated chord length is longer than the arc length, check to see if there was a mix-up with the units. The most common case is an angle that should have been interpreted as degrees but was instead treated as radians. An arc connecting two identical endpoints can never be shorter than the chord.