Angle conversion explained
Angles appear everywhere: in geometry and trigonometry, navigation and surveying, astronomy and artillery, drafting and computer graphics. This guide shows you how to convert between the most common angle units—degrees, radians, gradians, turns—and helpful sub‑units such as arcminutes, arcseconds, milliradians, and right‑ascension hours. You’ll also learn the key formulas, see common values, and work through practical examples.
Use the Angle Conversion Calculator above to translate any value instantly across units. Then come back here to understand the “why” and “how,” or to cite formulas in your work.
What is an angle?
An angle is formed by two rays (or line segments) sharing a common endpoint called the vertex. Angles measure “rotation” or “opening” between those rays.
We often classify angles by their size:
| Type | Degrees | Radians | Turns | Gradians |
|---|---|---|---|---|
| Acute | 0°–90° | 0–π/2 | 0–1/4 | 0g–100g |
| Right | 90° | π/2 | 1/4 | 100g |
| Obtuse | 90°–180° | π/2–π | 1/4–1/2 | 100g–200g |
| Straight | 180° | π | 1/2 | 200g |
| Reflex | 180°–360° | π–2π | 1/2–1 | 200g–400g |
| Full | 360° | 2π | 1 | 400g |
Related concepts you might encounter:
- Complementary angles: two angles summing to 90° (π/2).
- Supplementary angles: two angles summing to 180° (π).
- Reference angle: an acute angle “representing” any other angle’s distance to the nearest x‑axis.
- Central angle: an angle whose vertex is at a circle’s center, subtending an arc on the circumference.
Radians and the degree↔radian formulas
Radians are the natural mathematical unit for angles because they relate angle directly to arc length on a circle. One radian is the angle that subtends an arc equal to the circle’s radius (R).
- A full turn corresponds to a circumference (2\pi R) ⇒ 1 turn = (2\pi) radians = 360°.
- Therefore, 180° = (\pi) radians.
This gives the core conversion formulas:
Inline:
- Degrees → radians:
- Radians → degrees:
Display:
Common angles:
| Degrees | Radians |
|---|---|
| 15° | ( \pi/12 ) |
| 30° | ( \pi/6 ) |
| 45° | ( \pi/4 ) |
| 60° | ( \pi/3 ) |
| 90° | ( \pi/2 ) |
| 180° | ( \pi ) |
| 270° | ( 3\pi/2 ) |
| 360° | ( 2\pi ) |
Example: convert 225° to radians
Degrees–minutes–seconds (DMS) and decimal degrees
In navigation, surveying, and mapping you’ll often see degrees split into minutes of arc and seconds of arc:
- (1^\circ = 60′)
- (1′ = 60″)
- So (1^\circ = 60′ = 3600″)
To convert DMS → decimal degrees:
Worked example: (51^\circ 28′ 40″)
To convert decimal degrees → DMS:
- (d = \lfloor \text{deg}_\text{decimal} \rfloor)
- (m = \lfloor (\text{deg}_\text{decimal} - d) \times 60 \rfloor)
- (s = ((\text{deg}_\text{decimal} - d) \times 60 - m) \times 60)
Example: (23.5083^\circ \approx 23^\circ 30′ 29.88″) (round as needed).
Other useful angle units
Here are several units you’ll encounter and how to convert them. In our calculator, everything is first normalized to degrees and then rescaled to your target unit.
Turns (revolutions)
- Definition: 1 turn = a full circle.
- Relations: 1 turn = 360° = (2\pi) rad = 400g.
- Conversions:
- Degrees → turns: ( \text{turn} = \text{deg} / 360 )
- Radians → turns: ( \text{turn} = \text{rad} / (2\pi) )
Gradians (gon)
- Definition: a right angle is exactly 100 gradians ⇒ 1 full turn is 400g.
- Relation to degrees: 1g = 0.9° and 1° = (10/9)g.
- Conversions:
- Degrees → gradians: ( \text{g} = \tfrac{10}{9},\text{deg} )
- Radians → gradians: ( \text{g} = \tfrac{200}{\pi},\text{rad} )
- Turns → gradians: ( \text{g} = 400 \times \text{turn} )
Arcminutes and arcseconds
- Sub‑units of a degree used in geodesy and astronomy.
- Relations:
- (1^\circ = 60′), (1′ = 60″)
- In degrees: (1′ = 1/60^\circ), (1″ = 1/3600^\circ)
Milliradian (mrad) and microradian (µrad)
- Metric angular sub‑units useful for instrumentation and small‑angle approximations.
- Relations:
- (1,\text{rad} = 1000,\text{mrad})
- (1,\text{mrad} \approx 0.05729578^\circ)
- (1,\mu\text{rad} \approx 0.00005729578^\circ)
Artillery mils (two standards)
- Different “mil” definitions are used:
- NATO mil (6400‑circle): (1,\text{mil} = 360^\circ / 6400 = 0.05625^\circ)
- Soviet mil (6000‑circle): (1,\text{mil} = 360^\circ / 6000 = 0.06^\circ)
Choose the correct mil standard for your application; the calculator supports both.
Right ascension hours (astronomy)
- Celestial coordinates often use hours (h) around a 24‑hour circle.
- Relation: 1h = 15° and 24h = 360°.
How to use this angle converter effectively
- Enter your angle value and select its current unit (e.g., degrees, radians, gradians).
- Review converted values across grouped unit “cards” (core math units, surveying, precision, navigation).
- Adjust decimal precision in your workflow as needed; the converter displays practical defaults per unit.
- For citation or hand‑calculation, use the formulas below to show the steps.
Worked examples
- Degrees → radians
Given (60^\circ), compute radians:
- Radians → degrees
Given ( \tfrac{3\pi}{4} ) rad, compute degrees:
- DMS → decimal degrees
Convert (51^\circ 28′ 40″):
- NATO mils → degrees
Convert (1200) mils (6400‑circle):
References and further reading
- Angle — Wikipedia:
https://en.wikipedia.org/wiki/Angle - Radian — Wikipedia:
https://en.wikipedia.org/wiki/Radian - Degree (angle) — NIST Guide to the SI:
https://physics.nist.gov/cuu/Units/units.html - Conversion of units (Angle section) — Wikipedia:
https://en.wikipedia.org/wiki/Conversion_of_units#Angle - Gradian — Wikipedia:
https://en.wikipedia.org/wiki/Gradian
Tip: When writing or citing, prefer radians for calculus and most mathematical derivations, degrees for everyday descriptions and education, gradians for surveying contexts, and mils for artillery/fire control contexts.