Angle Unit Converter: Degrees, Radians, Grads, Mils & SR

Overview

Angle units look similar on paper, but they measure two different things: rotation in a plane versus spread in 3D space. Mixing those categories is the fastest way to get a “reasonable-looking” number that is completely wrong.

Plane angles describe turning: degrees (°), radians (rad), gradians (gon), arcminutes (′), arcseconds (″), and mils. Solid angles describe how much of the surrounding sphere a shape covers: steradians (sr), square degrees (sq°), and spats.

Where each family shows up

• Math/physics: radians and steradians dominate formulas and models (see Energy tools for more physics-oriented calculators).
• Surveying/GIS: degrees–minutes–seconds (DMS) and gradians are common (browse Home & DIY tools for measurement workflows).
• Astronomy & imaging: arcminutes/arcseconds and square degrees are standard for sky coverage.
• 3D graphics & rendering: solid angle connects directly to light distribution and field coverage (see Data tools for related computation utilities).

Quick reference landmarks (helps catch mistakes)

If your conversion makes a right angle come out far from 90°, π/2, or 100 gon, it’s almost always a unit-family mix-up or a wrong mil standard.

More unit tools live in Unit Converters.

Formula/Methodology

Reliable conversion starts by mapping every unit to a single base unit: radians for plane angles and steradians for solid angles. Then convert from the base to the target.

Plane angle base (radian):
θ(rad) = θ(from) × k_from
θ(to) = θ(rad) ÷ k_to

Key plane relationships:
rad = deg × (π/180)
deg = rad × (180/π)
gon = deg × (400/360) = deg × (10/9)
deg = gon × (360/400) = gon × (9/10)
arcmin = deg × 60
arcsec = deg × 3600

For solid angles, the definition ties directly to area on a sphere: Ω = A / r². On the unit sphere (r = 1), steradians numerically equal the surface area they cut out.

Solid angle base (steradian):
Ω(sr) = Ω(from) × k_from
Ω(to) = Ω(sr) ÷ k_to

Square degrees ↔ steradians:
1 sr = (180/π)² sq°
1 sq° = (π/180)² sr

Sphere totals:
Full sphere = 4π sr = 1 spat ≈ 12.56637 sr
Full sphere ≈ 41252.96125 sq°

Common math/logic mistakes (and how to spot them)

• Forgetting the square in solid angles: converting sq° ↔ sr uses (π/180)², not (π/180). If you’re off by ~57×, you likely missed the square.
• Treating “mil” as universal: “mil” depends on the standard (NATO 6400 per circle is common). If your reference uses a different circle division, results will disagree by a few percent.
• Mixing arcminutes with time minutes: arcminutes are 1/60 of a degree, not “minutes of time.” The symbols look similar in some contexts.
• Assuming wrapping rules are part of the unit: negative angles and angles above 360° can be valid; wrapping (mod 360°) is a convention you apply after converting.

Examples

1️⃣ 45° to radians

Use rad = deg × (π/180).

45 × π/180 = π/4 ≈ 0.785398 rad.

2️⃣ 200 gon to degrees

A full circle is 400 gon = 360°, so multiply by 0.9.

200 gon × 0.9 = 180°.

3️⃣ 3° 30′ to degrees (DMS step that’s often missed)

Convert minutes as base-60: 30′ = 30/60° = 0.5°.

So 3° 30′ = 3.5° (then convert to radians if needed).

4️⃣ 7200″ (arcseconds) to degrees

1° = 3600″, so divide by 3600.

7200″ ÷ 3600 = 2°.

5️⃣ 2 sr to square degrees

Use sq° = sr × (180/π)².

2 × (180/π)² ≈ 6565.6127 sq°.

Infographic & Visual Guide

Plane angle is “how far you turned.” Solid angle is “how much of the sphere you cover.” The key mental model: steradians measure area on the unit sphere, so they scale with a squared factor rather than linearly with degrees.

Field-of-view note (important edge case)

If you only know a single “FOV in degrees,” you usually cannot convert it to steradians without geometry. A cone, a rectangle, and an ellipse with the same “angle” can cover different solid angles.

• Circular cone (full angle θ): Ω = 2π(1 − cos(θ/2)).
• Rectangular sensor: you need both horizontal and vertical FOV (or sensor size + focal length) to estimate Ω.

Use Cases

• Surveying & construction layout: gradians (gon) simplify decimal subdivisions; DMS is common for bearings and GIS handoffs.
• Astronomy and imaging: arcminutes and arcseconds describe tiny separations; square degrees describe sky survey coverage.
• Military ranging/aiming: mil values are used in reticles and angular corrections; always confirm which mil definition your reference uses.
• Lighting and radiometry: steradians connect to intensity distribution and coverage in 3D models.
• Math and simulation: radians reduce errors in trig formulas and numerical methods when derivatives are involved.

Typical ranges (useful for sanity checks)

• Plane angles: common wrap is 0–360° (0–2π rad), but negatives and values above one turn can be meaningful.
• Solid angles: physically 0–4π sr for a closed sphere around a point; values above 4π indicate a modeling or unit mistake.

FAQ

Last review: December 2025