CalcDuck

Angle Converter

Angle Converter

Converted value
3.141593
Conversion factor
1 deg = 0.017453293 rad

A straight angle of 180 degrees equals exactly π radians, and a right angle of 90 degrees equals exactly 100 gradians, since a full circle is 360 degrees, 2π radians, 400 gradians or 1 turn. Every angle unit here converts through radians, the SI unit of angle, so any pair reduces to one exact ratio.

Tip: “Copy with settings” shares a link that opens this calculator with your numbers already filled in.

How this calculator works

This converter covers the angle units used in mathematics, engineering, surveying, astronomy and navigation: degrees, radians, gradians, arcminutes, arcseconds and turns (full revolutions).

Every unit is defined as a fraction of the radian, the SI unit of angle equal to the angle subtended when an arc's length equals the circle's radius. Because a full circle is exactly 2π radians, 360 degrees, 400 gradians or 1 turn, all four systems are exact rational or π-based multiples of each other with no approximation needed.

How it converts

result = value × (radians per from-unit) ÷ (radians per to-unit)Exact anchors: 1 full circle = 360° = 2π rad = 400 grad = 1 turn · 1° = 60′ = 3600″

Worked example: 90° to gradians

  1. 90° × (π/180 rad per degree) = π/2 rad.
  2. π/2 rad ÷ (π/200 rad per gradian) = 100 grad.
  3. So a right angle (90°) is exactly 100 gradians — the gradian system was designed to make a right angle a round number.

Frequently asked questions

Why do gradians exist alongside degrees?

The gradian (also called gon) was introduced during the French metric system reform to make a right angle equal to a clean 100 units, dividing the full circle into 400. It is still used in some surveying and civil engineering software.

When should I use radians instead of degrees?

Use radians for calculus and most programming math libraries (trigonometric functions in JavaScript, Python and similar languages expect radians), since radians make derivative and integral formulas for sine and cosine come out without extra conversion constants.

What are arcminutes and arcseconds used for?

They subdivide the degree for high precision: 1 degree = 60 arcminutes = 3,600 arcseconds. They appear in geographic coordinates (latitude and longitude), astronomy (apparent size of celestial objects) and optics.

What is a 'turn'?

A turn (also called a revolution or cycle) is one full rotation, equal to 360° or 2π radians. It is a convenient unit when reasoning about rotational speed or how many complete spins something makes.

How do I convert radians to degrees without a calculator?

Multiply the radian value by 180/π (about 57.2958). This follows directly from the exact anchor that a full circle is 2π radians = 360°, so 1 radian equals 180/π degrees. For example, 2 radians × 180/π ≈ 114.59°.

Sources

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