Angle Conversion: Degrees, Radians, and Gradians Introduction Understanding how to convert between degrees, radians, and gradians is a basic but essential skill across mathematics, science, engineering, surveying, and everyday measurement tasks. Whether you are writing code that uses trigonometric functions, interpreting survey bearings, or solving a physics problem, accurate angle conversion avoids errors and misinterpretations. This article covers the core definitions, the conversion formulas, step-by-step worked examples, a handy reference table, common real-world applications, and practical tips and best practices. The content is optimized for the keyword "angle conversion degrees radians gradians" and aims to be a concise yet comprehensive learning resource. Key Concepts / Definitions Degree (°): A circle is divided into 360 degrees. Degrees are the most familiar unit for everyday angles (compass headings, simple geometry). Radian (rad): A radian is defined from arc length: one radian is the angle subtended at the center of a circle by an arc equal in length to the circle's radius. A full circle equals 2π radians. Gradian (gon, grad): Also called gon or grad, a full circle equals 400 gradians. Gradians are used in surveying and in some national measurement systems; 100 gradians equals a right angle. Key equivalences: 1 revolution = 360° = 2π rad = 400 grad Therefore: 1° = π / 180 rad 1 grad = π / 200 rad 1 grad = 0.9° (so a grad is slightly smaller than a degree) Terminology: Gradians are sometimes abbreviated as gon or grad. Use the correct symbol: degrees (°), radians (rad), gradians (gon or grad). Conversion Formulas Here are the primary conversion formulas you will use for angle conversion degrees radians gradians. I highlight the most important ones in bold. Between degrees and radians: rad = deg × π / 180 deg = rad × 180 / π Between degrees and gradians: grad = deg × 10 / 9 deg = grad × 9 / 10 (Because 360° = 400 grad → 1° = 400/360 = 10/9 grad) Between radians and gradians: rad = grad × π / 200 grad = rad × 200 / π Practical constants (approximate): π ≈ 3.141592653589793 1° ≈ 0.0174532925199433 rad 1 rad ≈ 57.29577951308232° 1 grad = 0.9° 1° ≈ 1.111111111111111 grad Normalization: To normalize an angle to a principal interval (common in programming and engineering): To [0, 360): deg_normalized = deg mod 360 To (-180, 180]: map values >180 by subtracting 360, etc. Use analogous modular arithmetic in radians or gradians. Practical Examples (with worked calculations) Below are step-by-step worked conversions using the formulas above. Each example shows both the exact symbolic result (when possible) and a decimal approximation. Example 1 — Convert 30° to radians and gradians To radians: rad = deg × π / 180 rad = 30 × π / 180 = (30/180)π = (1/6)π Exact: π/6 rad Decimal: π/6 ≈ 3.141592653589793 / 6 ≈ 0.5235987756 rad To gradians: grad = deg × 10 / 9 grad = 30 × 10 / 9 = 300 / 9 = 33.333333... grad Decimal: 33.333333 grad Example 2 — Convert 135° to radians and gradians To radians: rad = 135 × π / 180 = (135/180)π = (3/4)π Exact: 3π/4 rad Decimal: 3π/4 ≈ 2.356194490192345 ≈ 2.35619449 rad To gradians: grad = 135 × 10 / 9 = 1350 / 9 = 150 grad Example 3 — Convert 2 radians to degrees and gradians To degrees: deg = rad × 180 / π deg = 2 × 180 / π = 360 / π ≈ 360 / 3.141592653589793 ≈ 114.5915590° To gradians (direct or via degrees): Direct: grad = rad × 200 / π grad = 2 × 200 / π = 400 / π ≈ 400 / 3.141592653589793 ≈ 127.3239545 grad Via degrees: convert 114.5915590° × 10 / 9 ≈ 127.3239544 grad (same within rounding) Example 4 — Convert 250 gradians to degrees and radians To degrees: deg = grad × 9 / 10 deg = 250 × 9 / 10 = 225° To radians: rad = deg × π / 180 = 225 × π / 180 = (225/180)π = (5/4)π Exact: 5π/4 rad Decimal: 5π/4 ≈ 3.926990716 Example 5 — Convert decimal degrees to DMS and back (useful for surveying) Convert 25.7625° to degrees-minutes-seconds (DMS): Degrees = 25° Fractional part = 0.7625° × 60 = 45.75' → minutes = 45' Fractional minutes = 0.75' × 60 = 45" → seconds = 45" Result: 25° 45' 45" Convert 25° 45' 45" back to decimal degrees: deg = 25 + 45/60 + 45/3600 = 25 + 0.75 + 0.0125 = 25.7625° Example 6 — Using trig libraries in code (reminder) Many programming languages expect radians for trig functions: Python: math.sin(math.radians(30)) converts 30° to radians internally. To compute sin(30°) correctly: sin(30°) = sin(π/6) = 0.5 If you pass degrees directly to sin in these languages, you will get the wrong result; always convert to radians first. Conversion Reference Table Below is a quick reference table of common angles with exact and approximate conversions. Use it when you need a quick lookup. | Degrees (°) | Radians (exact) | Radians (approx) | Gradians (grad) | |---:|---|---:|---:| | 0° | 0 | 0.0000000000 | 0 grad | | 30° | π/6 | 0.5235987756 | 33.333333 grad | | 45° | π/4 | 0.7853981634 | 50 grad | | 60° | π/3 | 1.0471975512 | 66.666667 grad | | 90° | π/2 | 1.5707963268 | 100 grad