Angle conversion formulas (with intuition)

Angles are one of those “looks simple, causes chaos” topics — because the math itself is easy, but mixing units can quietly break your answer. This page is designed to make conversions feel automatic. Here’s the key idea:

A full circle is the same physical rotation, no matter what unit you use. Different units are just different ways of labeling that same rotation. So every conversion can be done by comparing to a circle or a half‑circle.

• Degrees (°): The classic classroom unit. A full circle is 360°.
• Radians (rad): The “math-native” unit. A full circle is 2π rad. Radians connect angles to arc length: arc length = radius × angle (in radians).
• Gradians / gons (gon): Used in some surveying/engineering contexts. A full circle is 400 gon. A right angle is a clean 100 gon.
• Turns (rev): A full circle is 1 turn. Great for rotations, motors, and “how many spins?” thinking.

The easiest anchor is the half‑circle: 180° = π rad = 200 gon = 0.5 turn. Once you accept that, the conversion formulas fall out naturally:

• Degrees → Radians: rad = deg × (π / 180)
• Radians → Degrees: deg = rad × (180 / π)
• Degrees → Gradians: gon = deg × (10 / 9) (because 360° maps to 400 gon)
• Gradians → Degrees: deg = gon × (9 / 10)
• Degrees → Turns: turn = deg / 360
• Turns → Degrees: deg = turn × 360

From there, everything else can route through degrees (that’s how this calculator works internally): take your input unit → convert to degrees → convert degrees to all the other units. This is reliable because degrees are easy to read and it avoids having to memorize every possible “unit‑to‑unit” formula.

DMS looks scary until you remember it’s just base‑60. The formula is: decimal degrees = degrees + minutes/60 + seconds/3600. If the angle is negative, the minutes and seconds follow the sign of the degrees (this tool handles it for you).

Example: 30° 15′ 50″ becomes 30 + 15/60 + 50/3600 = 30.263888…°. Once you have decimal degrees, you can convert to radians, gradians, or turns normally.

Worked examples you can copy

Use rad = deg × (π / 180). So 45 × (π/180) = π/4 ≈ 0.785398… rad. In this tool, type 45, choose Degrees, and you’ll see radians plus everything else.

Radians → degrees uses deg = rad × (180 / π). If rad = π/3, then (π/3) × (180/π) = 60°. In the converter: type pi/3, choose Radians.

Since 400 gon is a circle and 360° is a circle, the ratio is deg = gon × (9/10). So 100 × 0.9 = 90°. That’s why surveying folks like gradians: right angles are a clean 100.

Turns are simple: deg = turn × 360, so 0.2 × 360 = 72°. Then to radians: 72 × (π/180) = 2π/5 ≈ 1.256637… rad.

Convert 12° 30′ 0″: 12 + 30/60 + 0/3600 = 12.5°. That’s an easy one because 30 minutes is exactly half a degree.

What this converter does behind the scenes

This calculator uses a “single source of truth” method: everything is converted to decimal degrees first, then degrees are converted to the other formats. That sounds boring, but it’s exactly why it’s reliable.

• If your unit is degrees/gradians/turns, the tool reads your number as a standard decimal (supports scientific notation like 1e-3).
• If your unit is radians, the tool also supports expressions containing π (like pi/6 or 2*pi) so you don’t have to pre-calculate.
• If your unit is DMS, the tool accepts flexible formats such as 30° 15' 50", 30 15 50, or even 30:15:50.

Once an input number is understood, it becomes degrees using the correct ratio: radians use deg = rad × 180/π, gradians use deg = gon × 9/10, turns use deg = turn × 360, and DMS uses deg + min/60 + sec/3600.

The tool then computes radians, gradians, turns, and a normalized DMS output. Finally it formats results using your rounding preference: Auto keeps simple answers clean (like showing 60° as π/3 when possible would be nice, but to keep it predictable we stick to decimals and show π-friendly input support instead).

If you’re working with friends, class, or a group chat, the Share buttons create a clean text summary you can paste. “Save Result” stores the last 20 conversions on this device, which is great for multi-step homework or comparing multiple angles quickly.

Frequently Asked Questions

• What’s the quickest way to convert degrees to radians?

  Multiply by π/180. If you only need common angles, memorize: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2.

• Why do math and physics prefer radians?

  Because radians connect angles to geometry naturally. In radians, arc length is s = r·θ and many formulas simplify. For example, the derivative of sin(x) is cos(x) only when x is in radians.

• What are gradians (gons) used for?

  Gradians are common in some surveying and engineering contexts because a right angle is exactly 100 gon. That makes quarter turns and right angles feel “base‑10 friendly.”

• What does “turns” mean?

  Turns (revolutions) measure rotation as a fraction of a full circle. 1 turn is one full rotation. It’s great for motors, wheels, spins, and oscillations.

• How do I type DMS on a normal keyboard?

  You can type 30 15 50 (spaces), 30:15:50 (colons), or 30 15 (degrees and minutes only). The converter will treat missing seconds as 0.

• Does this converter handle negative angles?

  Yes. Negative angles are fully supported across all units (including DMS). The output DMS will keep the sign on the degrees portion.

• Why do my answers have tiny rounding differences?

  Because π is irrational (its decimal never ends). Any decimal representation must be rounded. For most homework and everyday engineering use, 6 decimals is already very precise.