In geometry, a degree is the unit of angle measurement, equal to 1/360 of a circle. When measuring angles from the centre of circle, degrees can also be expressed as radians. The radian measure of an angle is the ratio between the length of the arc opposite the angle and the radius of the circle. An arc is a portion of the circumference of a circle, so the arc length is defined as the length of that portion. By converting an angle’s degree measure to radian measure, you can find the arc length opposite a given angle, measured in a unit of length such as inches or millimetres.

Convert a given angle’s degree measure to radians. The formula for converting degrees to radians is:

radians=degrees x pi/180.

Multiply your degree measure by pi/180. For instance, to convert 50 degrees to radians, multiply 50 by pi/180. This equals 50(pi)/180, which reduces to 5(pi)/18.

Determine the radius of the circle on which the angle measurement exists. You need to know the radius of the circle to calculate the arc length of the angle.

Convert the radian measure from step 1 to arc length. Because radian measure is the ratio of arc length to radius, you can calculate the arc length of an angle by multiplying the radian measure by the radius of the circle. Mathematically, this is expressed as:

radians = s/r, where s equals arc length and r equals radius. Therefore, arc length is s = (radians) times r.

In the example from step 1, the degree measure was 50, which equals 5(pi)/18 radians. To determine the angle’s corresponding arc length, multiply 5(pi)/18 by the radius you determined in step 2. If the radius is 10 inches, the arc length equals 10 x 5(pi)/18, or 50(pi)/18.

Convert the result to decimal form using your calculator. Use the approximation pi = 3.14159. In the example, enter 50 x 3.14159 on your calculator. Press ÷, then 18. Press equals to get the result, which rounds to 8.727 inches. Because the radius was expressed in inches, the arc length is also expressed in inches.

Convert the inches measurement to millimetres by multiplying the result by 25.4. So, 8.727 in equals 221.666mm.