It only makes sense to convert an angle (ø) to a distance (d) when the distance in question is on the circumference of a circle or on the surface of a sphere. When that is the case, use the equation ø = d/r – where r is the radius of the circle or sphere. This gives a value in radians, which is easy to convert to degrees. If you know the angle in degrees and want to find the arc length, convert the angle to radians and then use the converse expression: d = ø • r. To get distance in English units, you must express the radius in English units. Similarly, you must express the radius in metric units to get the distance in kilometers, meters, centimeters or millimeters.
Measuring Angles in Radians
A radian is an angular measurement based on the length of the radius of a circle or sphere. The radius is a line drawn from the center of the circle to a point A on its circumference or on its perimeter if it is a sphere. When a radial line moves from point A to another point B on the circumference, it traces an arc of length d while, at the same time, scribing an angle ø at the center point of the circle.
By definition, one radian is the angle you scribe when the length of the arc from point A to point B equals the length of the radius. In general, you determine the magnitude of any angle ø in radians by dividing the arc length traced by the radian lines between two points by the radius. This is the mathematical expression: ø (radians) = d/r. For this expression to work, you must express arc length and radius in the same units.
For example, suppose you want to determine the angle of the arc traced by radial lines extending from the center of the earth to San Francisco and to New York. These two cities are 2,572 miles (4,139 kilometers) apart, and the equatorial radius of the earth is 3,963 miles (6378 kilometers). We can find the angle using either the metric or English units, as long as we use them consistently: 2,572 miles/3,963 miles = 4,139 km/6,378 km = 0.649 radians.
Radians to Degrees
We can derive a simple factor to convert from radians to degrees by noting that a circle has 360 degrees and that the circumference of the circle is 2πr units in length. When a radial line traces an entire circle, the arc length is 2πr/r = 2π, and since the line traces an angle of 360 degrees, we know that 360 degrees = 2π radians. Dividing both sides of this equation by 2, we get:
- 180 degrees = π radians
This means that 1 degree = π/180 radians and 1 radian = 180/π degrees.
Converting Degrees to Arc Length
We need one key piece of information before we can convert degrees to arc length, and that's the radius of the circle or sphere on which we measure the arc. Once we know it, the conversion is simple. Here's the two-step procedure:
- Convert degrees to radians.
- Multiply by the radius to get the arc length in the same units.
If you know the radius in inches and you want the arc length in millimeters, you must first convert the radius to millimeters.
A 50-Inch Circle Example
In this example, you want to determine the length of the arc – in millimeters – on the circumference of a circle with a diameter of 50 inches traced by a pair of lines that form an angle of 30 degrees.
- Start by converting the angle to radians. 30 degrees = 30π/180 radians. Since π equals approximately 3.14, we get 0.523 radians.
- Remember that the radius of a circle is half its diameter. In this case, r = 25 inches.
- Convert the radius to the target units – millimeters – using the conversion 1 inch = 25.4 millimeters. We get 25 inches = 635 millimeters.
- Multiply the radius by the angle in radians to get the arc length. 635mm • 0.523 radians = 332.1 mm.