AkCalculators

Angle conversion explained — degrees, radians, grads, arcminutes and slope

Angles are everywhere: from navigation and surveying to programming graphics and structural engineering. While the mathematics of angles is straightforward, the variety of units (degrees, radians, grads, arcminutes, arcseconds, milliradians, mils, and percent slope) can cause confusion. This Angle Converter provides precise conversions and the background to understand when and why each unit is used.

Why radians are the canonical unit

Mathematicians and many engineering formulas prefer radians because they make calculus and trigonometry simple: arc length = radius × angle (when angle is in radians). Most conversion flows use radians as the internal canonical unit because every other unit can be computed directly from radians using exact multipliers.

Common angle units and definitions

• Degree (°) — 1 full circle = 360°. Common in navigation, construction, and everyday usage.
• Radian (rad) — 1 full circle = 2π radians. 1 rad ≈ 57.2958°. Radians are natural for mathematical operations.
• Grad / Gon — 1 full circle = 400 grads (gons). 1 grad = 0.9°. Used historically in surveying; still used in some surveying equipment.
• Turn / Revolution — 1 turn = 360° = 2π rad. Useful for expressing fractions of full rotations.
• Arcminute (') and Arcsecond (") — subdivisions of degrees: 1' = 1/60°; 1" = 1/3600°. Common in astronomy and precise geodesy.
• Milliradian (mrad) — 1 mrad = 0.001 rad. Common in ballistics, optics, and engineering (also used for angular measurements in lasers and scopes).
• Mil (1/6400 turn) — many military and artillery contexts use a circle divided into 6400 mils (note: variants exist, e.g., 6000 or 6283; this converter uses the common 6400 definition). 1 mil ≈ 0.05625°.
• Percent slope (%) — Represents the tangent of an angle expressed as a percentage: percent = tan(angle) × 100. Used in road grades, ramp specifications, and civil engineering.

Formulas used by this converter

Conversions go through radians internally. Key formulas:

• Degrees ↔ Radians: rad = deg × π / 180 ; deg = rad × 180 / π
• Grads (gons): grad = deg × 10/9 ; deg = grad × 9/10
• Turns: turn = deg / 360 ; deg = turn × 360
• Arcminutes & Arcseconds: arcmin = deg × 60 ; arcsec = deg × 3600 (and inversely)
• Milliradian: mrad = rad × 1000 ; rad = mrad / 1000
• Mil (1/6400 turn): mil = turn × 6400 ; turn = mil / 6400
• Percent slope: percent = tan(rad) × 100 ; rad = atan(percent / 100)

Worked examples

Example 1 — Degrees to radians: Convert 45° to radians.

45 × π / 180 = π / 4 ≈ 0.785398 rad.

Example 2 — Radians to arcseconds: Convert 0.0001 rad to arcseconds.

deg = 0.0001 × 180 / π ≈ 0.0057296°. arcsec = 0.0057296 × 3600 ≈ 20.6266″.

Example 3 — Percent slope to degrees: A road has a 12% grade. What is the angle in degrees?

rad = atan(12 / 100) ≈ atan(0.12) ≈ 0.11942 rad. deg ≈ 0.11942 × 180/π ≈ 6.842°.

Example 4 — Mils to degrees: 100 mil (1/6400 turn) → degrees: turn = 100/6400 = 0.015625 turn; deg = 0.015625 × 360 = 5.625°.

When to use which unit

• Degrees: Everyday measurements, navigation, CAD interfaces, and most user-facing displays.
• Radians: Mathematical modeling, calculus, simulations, and many programming libraries expect radians.
• Grads: Surveying equipment and legacy civil-engineering documents sometimes use grads.
• Arcminutes/Arcseconds: Astronomy, geodesy, and telescope pointing require arcminute/arcsecond precision.
• Milliradians & Mils: Ballistics, optical scopes, and artillery use these for small-angle approximations and quick aiming corrections.
• Percent slope: Road/rail/cycleway design and accessibility ramps — designers often specify maximum percent grades rather than degrees.

Best practices and precision tips

• Keep calculations in radians when chaining trigonometric functions — this avoids unit mismatch bugs in code.
• Mind small-angle approximations: For very small angles (in radians), sin(θ) ≈ θ and tan(θ) ≈ θ — useful for back-of-envelope estimates but not for precision work.
• Be explicit about mil definition: If exchanging values with military or international partners, confirm which mil system they use (this converter documents the 1/6400 version).
• Slope vs angle: When specifying ramps, percent can be more intuitive; when modeling forces, use radians or degrees as appropriate.

Common mistakes to avoid

• Feeding degrees into trig functions in languages that expect radians (or vice versa) — this is a very common source of errors in code.
• Using percent slope interchangeably with percent of circle — percent slope is slope × 100, not a fraction of a circle.
• Assuming all mils are equal — clarify which definition (6400, 6250, 6000, or 6283) is expected when interoperability matters.

SEO-friendly notes & how to use this converter

This page helps users find quick answers for queries like “convert degrees to radians”, “degrees to arcminutes”, “percent slope to degrees”, and “mils to degrees”. The converter is fast, precise, and designed to be used in engineering, surveying, programming, and educational contexts.

FAQ

Conclusion

Whether you’re coding in a language that expects radians, specifying a road grade as a percent, or aiming with a scope in mils, this Angle Converter gives accurate, reliable conversions. Keep it bookmarked for navigation, surveying, optics, engineering, and education tasks.