Angle Converter & Sexagesimal DMS Calculator

Convert planar angle measurements across Degrees, Radians, Gradians, Arcsec, NATO Mils, and Turns with live Sexagesimal DMS coordinates & subtense rangefinding.

Angle Value

From Unit

To Unit

Subtense Range Distance D (m)

Geometric, Military & Astronomical Milestones

🎯 Estimate-First Geometric Challenge

Before checking below, can you estimate how many Radians (rad) equal 180 °?

Target (Radians)

3.14159265

Sexagesimal DMS & Optical Subtense (W ≈ D · θ)

180° 0' 0"

Width W = 3,141.593 m @ 1000 m Range

Canonical geodetic coordinates & military subtense rangefinding

Simultaneous 14-Unit Equivalency Matrix

Degrees (°)

180

Radians (rad)

3.141593

Gradians (gon) (grad)

200

Arcminutes (')

10,800

Arcseconds (")

648,000

Milliarcseconds (mas)

6.480000e+8

Milliradians (SI) (mrad)

3,141.592654

NATO Artillery Mils (1/6400) (mil (NATO))

3,200

Soviet / Warsaw Mils (1/6000) (mil (USSR))

3,000

Swedish Mils (1/6300) (mil (SE))

3,150

Turns / Revolutions (turn)

0.5

Quadrants (Right Angle) (quad)

2

Sextants (60°) (sextant)

3

Zodiac Signs (30°) (sign)

6

Step-by-Step Mathematical Walkthrough

Step 1: Convert source unit to base SI Radians (rad)

Radians = 180 ° × 0.0174532925199 = 3.14159265 rad

Step 2: Convert Radians to target unit (Radians)

Radians = 3.14159265 rad / 1.00000000000 = 3.14159265 rad

Step 3: Sexagesimal DMS (Degrees° Minutes' Seconds") Decomposition

Decimal Degrees = 180.000000° → Integer D = 180°, Minutes M = 0', Seconds S = 0.0000"

Step 4: Optical Subtense Target Width (W ≈ D · θ_rad)

At Range D = 1,000 m and Angle θ = 3.141593 rad → Subtended Target Width W = 3,141.5927 m

About the Angle Converter

An angle converter is an essential tool in mathematics, geometry, physics, and mechanical engineering used to translate angular measurements between different units of circle division. Angles represent the amount of rotation between two intersecting lines and are fundamental to trigonometry, astronomy, navigation, and computer graphics calculations.

Mathematical Formula & Logic

The core mathematical conversions rely on the relation that a full circle (1 turn) is equivalent to 360 degrees, 2π radians, or 400 gradians.

- Degrees to Radians: Rad = Deg × (π / 180)
- Radians to Degrees: Deg = Rad × (180 / π)
- Degrees to Gradians: Grad = Deg × (400 / 360)
- Degrees to Turns: Turn = Deg / 360

Step-by-Step Example

Convert 90 degrees to Radians: 90° × (π / 180) = 90 × 0.01745329 = 1.570796 radians (which is exactly π / 2).

Reference Data & Values

unit	deg	rad	grad
1 Turn (Full Circle)	360°	2π rad (~6.283)	400 grad
Half Circle	180°	π rad (~3.142)	200 grad
Right Angle	90°	π/2 rad (~1.571)	100 grad
Standard Angle	45°	π/4 rad (~0.785)	50 grad

Frequently Asked Questions

What is a radian and why is it used in calculus?＋

A radian is the standard unit of angular measure defined by the arc length of a circle equal to its radius. It is preferred in calculus because it simplifies trigonometric derivatives and integrals, avoiding complex conversion factors.

Why are there 360 degrees in a circle?＋

The 360-degree circle originated from ancient Babylonian astronomers. They used a sexagesimal (base-60) numerical system and tracked the sun's movement, which was approximately 1 degree per day across a 360-day calendar cycle.

What is a gradian?＋

A gradian (or gon) is a unit of angle equal to 1/400 of a full circle. It was designed in France to metricate angular measurement, making right angles exactly 100 gradians for easier decimal calculations.