Logo

Angle & Solid Angle Converter

Convert between degrees, radians, gradians, mils, arcminutes, steradians, and more for geometry, surveying, and physics.

Enter Value & Units

(show mils, grads, arcsec … - Only Plane Angle)

Result

Converted: –

Scientific: –

Grouped: –

🔍

    Overview: Convert Plane and Solid Angles Easily

    The Angle & Solid Angle Converter is a versatile tool for converting between common plane angle units like degrees (°), radians (rad), gradians (gon), arcminutes (′), arcseconds (″), and mil (used in NATO, Sweden, and Warsaw Pact systems).

    It also supports solid angle conversions for spatial measurements used in 3D geometry and physics: steradians (sr), square degrees (deg²), and spats (entire sphere).

    Ideal for professionals in engineering, astronomy, military applications, surveying, and 3D rendering, this tool ensures accurate results across all systems. Simply input your value, choose the input/output units, and toggle between angle types. Results are delivered instantly and with scientific precision.

    Formula & Methodology: How Angle Conversions Work

    All conversions are routed through the SI base unit—radian (for plane) or steradian (for solid)—to ensure universal consistency. The general formulas are: 

    Plane Angles:
θrad = θinput × ffrom
θoutput = θrad ÷ fto

Solid Angles:
Ωsr = Ωinput × ffrom
Ωoutput = Ωsr ÷ fto

    Key angle reference values:

    • 1 full rotation = 360° = 2π rad = 400 grad
    • 1 degree = π/180 rad ≈ 0.01745 rad
    • 1 arcminute = 1/60°
    • 1 arcsecond = 1/3600°
    • 1 NATO mil = 2π / 6400 rad ≈ 0.0009817 rad
    • 1 steradian ≈ 3282.80635 deg²
    • 1 spat = 4π sr ≈ 12.566 sr (entire sphere)

    Examples: Real-World Angle Conversions

    • Convert 45° to radians:
      45 × π / 180 = 0.785398 rad
    • Convert 1 NATO mil to degrees:
      (2π / 6400) × (180 / π) ≈ 0.05625°
    • Convert 2 steradians to deg² (2 sr → deg²):
      2 × (180 / π)² ≈ 6573.0 deg²
    • Convert 200 gradians to degrees:
      200 × (360 / 400) = 180°
    • Convert 1 spat to steradians:
      1 spat = 4π sr ≈ 12.56637 sr

    Use Cases: Who Uses Angle & Solid Angle Conversions?

    • 🎯 Military & Artillery: Mils are used for aiming and targeting systems in NATO and other armed forces.
    • 📐 Civil Engineering & Surveying: Grads (gon) are used for precise angle division in leveling and boundary calculations.
    • 🧮 Mathematics & Trigonometry: Radians are essential for working with circular functions in formulas.
    • 🔭 Astronomy & Astrophysics: Steradians and square degrees describe star maps, fields of view, and sky coverage.
    • 🖥️ 3D Graphics & Lighting: Solid angles define light intensity distribution, falloff, and rendering volumes.
    • 🚧 Architecture: Degree and radian values help design angular layouts and projections.
    • 📚 Education & Exams: Geometry students regularly convert between degrees, radians, and minutes/seconds.

    Frequently Asked Questions

    What is a plane angle?

    A plane angle measures the rotation between two intersecting lines in a flat surface. Common units include degrees (°), radians (rad), and gradians (gon).

    What is a solid angle?

    A solid angle measures the three-dimensional spread of an object or shape from a single point. It is measured in steradians (sr), square degrees (deg²), or spats.

    What is the difference between degrees and gradians?

    A full circle equals 360 degrees or 400 gradians. Gradians simplify base-10 calculations, often used in surveying.

    Why do radians matter in mathematics?

    Radians are the natural unit of angular measurement in trigonometry and calculus because they simplify equations involving π and circle geometry.

    What is a NATO mil?

    A mil (NATO) is 1/6400th of a full circle, used in military rangefinding and targeting. It approximates 0.05625 degrees.

    How many steradians are in a full sphere?

    A full sphere equals 4π steradians, or approximately 12.56637 sr. One spat is equivalent to one full sphere.

    Can this converter switch between arcseconds and degrees?

    Yes. This tool can convert between degrees, arcminutes (′), and arcseconds (″), including fractional values and astronomy-specific inputs.

    What is a spat?

    A spat is a rarely used unit representing the solid angle of an entire sphere—equal to 4π steradians.

    Is the tool mobile-friendly?

    Yes! The converter is fully responsive, works offline in most browsers, and supports advanced and scientific units with no login required.