Angle Conversion: Degrees, Radians, Gradians, and Turns Understanding how to convert between different angle units is essential in engineering, physics, surveying, computer graphics, and many everyday tasks. This article explains the key concepts, gives the exact conversion formulas, walks through worked examples, and shows common applications and best practices to avoid errors when converting angles. Introduction Angles can be measured in several unit systems. The most familiar is the degree, but radians are the natural unit in calculus and physics, gradians (also called gons or grads) are used in surveying, and turns (revolutions) provide a convenient fractional view of a full circle. All these systems describe the same geometric quantity — the portion of a circle — so conversions are straightforward once you know the equivalences. This article is a practical reference: you will find step-by-step examples, exact formulas (using π where appropriate), a handy conversion table, and guidance on where each unit is commonly used. Key Concepts / Definitions Full circle (complete revolution): the entire 360-degree sweep around a point. This is the common reference for converting between units. Degree (°): 1 full circle = 360°. Historically divided into 360 parts. Radian (rad): based on arc length. 1 radian is the angle subtended by an arc equal in length to the circle's radius. 1 full circle = 2π radians. Gradian (gon or grad): a decimal-friendly subdivision. 1 full circle = 400 gradians. Therefore 1 grad = 0.9°. Turn (rev): a full revolution. 1 full circle = 1 turn. Fractional turns are often used (e.g., 0.25 turns = 90°). Fundamental equivalence: 360° = 2π rad = 400 grad = 1 turn From this follow the base conversion factors: 1° = π / 180 rad 1 rad = 180 / π ° 1 grad = 0.9° = 9/10 ° 1° = 10/9 grad 1 turn = 360° = 2π rad = 400 grad Keep in mind: calculators and programming libraries typically require angles in either degrees or radians — use the correct mode or convert as needed. Conversion Formulas Below are the essential formulas. I use bold to highlight the key formulas and conversion constants. Base full-circle identity: 360° = 2π rad = 400 grad = 1 turn Degree ↔ Radian: radians = degrees × (π / 180) degrees = radians × (180 / π) Degree ↔ Gradian: gradians = degrees × (10 / 9) (because 360° → 400 grad) degrees = gradians × (9 / 10) Turn ↔ Degree/Radian/Gradian: turns = degrees / 360 degrees = turns × 360 turns = radians / (2π) radians = turns × 2π turns = gradians / 400 gradians = turns × 400 Radian ↔ Gradian (derived): radians = gradians × (π / 200) (since 400 grad = 2π rad) gradians = radians × (200 / π) Exact constants to remember: π ≈ 3.141592653589793 1° = π/180 rad ≈ 0.0174532925199433 rad 1 rad = 180/π ° ≈ 57.2957795130823° 1 grad = 0.9° 1 turn = 360° Practical Examples (with worked calculations) Below are step-by-step conversions for common cases. I show the algebraic conversion first (exact form with π when appropriate) and then the decimal approximation. Example 1 — Convert 45° to radians: 1. Formula: radians = degrees × (π / 180) 2. Calculation: radians = 45 × (π / 180) = (45/180)π = (1/4)π 3. Decimal: (1/4)π ≈ 0.7853981633974483 rad Result: 45° = π/4 rad ≈ 0.785398 rad Example 2 — Convert 2.0 radians to degrees: 1. Formula: degrees = radians × (180 / π) 2. Calculation: degrees = 2.0 × (180 / π) = 360 / π 3. Decimal: 360 / π ≈ 114.59155902616465° Result: 2.0 rad ≈ 114.5916° Example 3 — Convert 250 gradians to degrees and radians: 1. Gradians → Degrees: degrees = gradians × (9 / 10) degrees = 250 × (9 / 10) = 225° 2. Degrees → Radians: radians = degrees × (π / 180) radians = 225 × (π / 180) = (225/180)π = (5/4)π decimal: (5/4)π ≈ 3.9269908169872414 rad Result: 250 grad = 225° = 5π/4 rad ≈ 3.926991 rad Example 4 — Convert 0.125 turns to degrees, radians, and gradians: 1. Turns → Degrees: degrees = turns × 360 degrees = 0.125 × 360 = 45° 2. Degrees → Radians: radians = degrees × (π / 180) radians = 45 × (π / 180) = π/4 ≈ 0.7853981634 rad 3. Degrees → Gradians: gradians = degrees × (10 / 9) gradians = 45 × (10 / 9) = 50 grad Result: 0.125 turns = 45° = π/4 rad ≈ 0.785398 rad = 50 grad Example 5 — Convert -30° (a negative angle) to radians: 1. radians = −30 × (π / 180) = −(1/6)π 2. decimal: −π/6 ≈ −0.5235987755982988 rad Result: −30° = −π/6 rad ≈ −0.523599 rad Example 6 — Convert 3 degrees 15 minutes (3°15') to radians: 1. Convert minutes to decimal degrees: 15' = 15/60 = 0.25° total degrees = 3 + 0.25 = 3.25° 2. radians = 3.25 × (π / 180) = (3.25π / 180) ≈ 0.056775 rad Result: 3°15' ≈ 0.056775 rad These examples demonstrate standard manipulations: convert to a common base (degrees or radians) then apply appropriate factors. Conversion Reference Table | Quantity | Degrees (°) | Radians (rad) | Gradians (grad / gon) | Turns | |---:|:---:|:---:|:---:|:---:| | Full circle | 360° | 2π | 400 grad | 1 turn | | Right angle | 90° | π/2 ≈ 1.570796 | 100 grad | 0.25 turn | | 45° | 45° | π/4 ≈ 0.785398 | 50 grad