terican

BIPM-ratified constants · v1.0

Converter

Gradians, to degrees converter calculator.

Convert gradians (gons) to degrees using degrees = gradians x 0.9. Supports both grad-to-degree and degree-to-grad conversion instantly.

From

gradians

grad_to_deg

100 grad_to_deg =90Degrees

Equivalents

Precision: 6 dp · Notation: Decimal · 2 units

→ Degrees

Gradiansgrad_to_deg90

→ Gradians

Degreesdeg_to_grad111.1111

Common pairings

1 grad_to_degequals1.1111 deg_to_grad
1 deg_to_gradequals0.9 grad_to_deg

The conversion

How the value
is computed.

Gradians to Degrees: Understanding Angle Unit Conversion

Angle measurement relies on several different unit systems depending on the field of application. Two of the most important are gradians (also called grads or gons) and degrees. Converting between these units is essential in surveying, civil engineering, military navigation, and scientific computing. This calculator handles both directions: grad to degree and degree to grad.

What Are Gradians?

A gradian (symbol: grad or gon) divides a full circle into exactly 400 equal parts. This makes the gradian a metric-friendly angle unit — right angles equal exactly 100 gradians and straight angles equal exactly 200 gradians. The gradian system emerged during the French Revolution alongside the metric system, designed to bring base-10 logic to angular measurement.

According to the BIPM SI Brochure, the gradian (gon) is a non-SI unit accepted for use alongside SI units in technical fields, particularly geodesy and surveying, due to its widespread adoption in professional instrumentation.

What Are Degrees?

A degree (symbol: °) divides a full circle into 360 equal parts. This system traces back to ancient Babylonian astronomy and remains the most widely used angle measurement system globally. Right angles equal 90 degrees; straight angles equal 180 degrees.

The Conversion Formula

Since a full circle contains 400 gradians and 360 degrees, the conversion ratio is exact and simple:

Degrees = Gradians × (9 / 10)

Or equivalently: Degrees = Gradians × 0.9

For the reverse conversion: Gradians = Degrees × (10 / 9)

This relationship derives directly from dividing one full-circle definition by the other: 360 ÷ 400 = 9/10 = 0.9. Every 10 gradians equals exactly 9 degrees — one of the cleanest conversion ratios in mathematics, requiring no approximation whatsoever.

Step-by-Step Conversion Examples

Example 1: 100 Gradians to Degrees

  • Start with: 100 gradians
  • Apply the formula: 100 × 0.9 = 90
  • Result: 90 degrees (a right angle)

Example 2: 200 Gradians to Degrees

  • Start with: 200 gradians
  • Apply the formula: 200 × 0.9 = 180
  • Result: 180 degrees (a straight angle)

Example 3: 50 Gradians to Degrees

  • Start with: 50 gradians
  • Apply the formula: 50 × 0.9 = 45
  • Result: 45 degrees

Example 4: 135 Degrees to Gradians (Reverse)

  • Start with: 135 degrees
  • Apply the reverse formula: 135 × (10/9) = 150
  • Result: 150 gradians

Key Reference Values

  • 0 grad = 0°
  • 100 grad = 90° (right angle)
  • 200 grad = 180° (straight angle)
  • 300 grad = 270°
  • 400 grad = 360° (full circle)
  • 1 grad = 0.9°
  • 1° = 1.1111... grad

Practical Applications

Surveying and Geodesy

Professional surveyors in continental Europe — particularly France, Germany, and the Netherlands — routinely use gradians. Many total stations and theodolites display angles in gons (gradians) by default. When exchanging data between teams using different instruments, accurate grad-to-degree conversion is critical to avoid positioning errors.

Civil Engineering and Slope Calculations

Slope calculations, road banking angles, and structural alignment frequently use gradians because the base-10 system simplifies decimal arithmetic. A slope of 1 gradian corresponds closely to a 1% grade, enabling straightforward mental calculation on-site.

Scientific Calculators and Computing

Most scientific calculators, including the TI-30XS MultiView, offer a GRAD mode alongside DEG and RAD modes. When GRAD mode is active, all trigonometric functions accept and return values in gradians. Understanding when a calculator is operating in GRAD mode — and how to convert results to degrees — is essential for avoiding silent computational errors.

Methodology and Sources

The conversion formula used in this calculator derives directly from the definitional relationship between the two angle units: 360 degrees and 400 gradians both represent exactly one complete revolution. This is documented in the NIST Guide for the Use of the International System of Units (SI), which provides authoritative standards for unit conversion methodology used in scientific and engineering practice. The BIPM SI Brochure further establishes the gradian (gon) as a recognized non-SI unit accepted for use alongside SI units in technical domains. The conversion factor 9/10 is exact by definition — no rounding or approximation is involved at any step.

Reference

Frequently asked questions

What is a gradian and how does it differ from a degree?
A gradian (also called a gon or grad) divides a full circle into 400 equal parts, while a degree divides it into 360 equal parts. This means 1 gradian equals 0.9 degrees and 1 degree equals approximately 1.1111 gradians. The gradian was developed during the French Revolution as part of a metric angle system, making right angles exactly 100 gradians instead of 90 degrees, which simplifies base-10 calculations.
What is the formula to convert gradians to degrees?
The formula is: degrees = gradians x (9/10), or equivalently degrees = gradians x 0.9. This exact factor derives from the ratio of 360 degrees to 400 gradians in a full circle (360 / 400 = 0.9). For example, 50 gradians equals 50 x 0.9 = 45 degrees, and 400 gradians equals exactly 360 degrees. No rounding is required — the conversion is exact.
How do you convert degrees to gradians?
To convert degrees to gradians, multiply the degree value by 10/9 (approximately 1.1111). The formula is: gradians = degrees x (10/9). For example, 90 degrees equals 90 x (10/9) = 100 gradians; 180 degrees equals 200 gradians; and 45 degrees equals 50 gradians. This reverse conversion is equally exact, producing integer results for most common angle benchmarks.
Why do scientific calculators have a GRAD mode?
Scientific calculators include a GRAD mode because gradians are used professionally in surveying, geodesy, and civil engineering, especially in continental Europe. Calculators such as the TI-30XS MultiView and HP 35s support GRAD alongside DEG and RAD modes. When GRAD mode is active, all trigonometric functions interpret input and output values in gradians rather than degrees or radians, making it essential to identify the active mode before performing calculations.
How many gradians equal a right angle, and why does that matter?
A right angle equals exactly 100 gradians. This is the primary practical advantage of the gradian system — it aligns angular measurement with base-10 arithmetic. One quarter circle = 100 grad, half circle = 200 grad, three-quarter circle = 300 grad, and a full circle = 400 grad. This regularity eliminates the irregular 90/180/270/360 sequence found in degrees, simplifying surveying calculations and reducing arithmetic errors in field conditions.
In which fields and countries are gradians most commonly used?
Gradians are most commonly used in professional surveying and geodesy, particularly in France, Germany, the Netherlands, and other continental European countries where metrication was adopted comprehensively. Theodolites and total stations used in land surveying frequently display readings in gons (gradians) by default. Civil engineering applications involving slope measurement, road design, and topographic mapping also use gradians when decimal-based precision is preferred over traditional degree notation.