Angular Measurement and Linear Conversion

Angular measurements, expressed in degrees (°), represent a portion of a circle's circumference. Linear measurements, such as millimeters (mm), quantify a straight-line distance. Conversion between these systems requires additional contextual information, specifically the radius of the arc or circle involved. The relationship between angular and linear measurements is fundamentally tied to the concept of arc length.

Arc Length Calculation

The arc length (s) of a circle segment is determined using the following formula:

s = rθ

Where:

• s represents the arc length (in mm if the radius is in mm).
• r represents the radius of the circle (in mm).
• θ represents the angle subtended by the arc (in radians).

Converting Degrees to Radians

Since the formula utilizes radians, degrees must first be converted. The conversion factor is:

180° = π radians

Therefore, to convert degrees to radians:

θ (radians) = θ (degrees) (π / 180°)

Complete Conversion Process

1. Determine the radius (r) of the circle in millimeters.
2. Convert the angle (θ) from degrees to radians using the formula above.
3. Substitute the values of r and θ (in radians) into the arc length formula: s = rθ
4. The result (s) represents the arc length in millimeters.

Example

To find the arc length of a circle with a radius of 100 mm and a central angle of 30°, first convert 30° to radians: 30° (π / 180°) ≈ 0.524 radians. Then, substitute into the formula: s = 100 mm 0.524 radians ≈ 52.4 mm.

Applications

This conversion is crucial in various fields, including engineering, surveying, and computer graphics, whenever calculations involving both angular and linear dimensions are necessary.