Gradian — Definition, Formula & Examples

A gradian (also called a gon or grad) is a unit of angle measurement where a full circle is divided into 400 equal parts. One gradian equals 0.9 degrees, so a right angle is exactly 100 gradians.

The gradian is an angular unit defined such that one complete revolution measures 400 gradians (400 gon). Equivalently, 1 gradian equals $\frac{\pi}{200}$ radians or $\frac{9}{10}$ of a degree.

Key Formula

G = D \times \frac{10}{9}

Where:

$G$ = Angle measured in gradians
$D$ = Angle measured in degrees

How It Works

Gradians were designed so that a right angle equals exactly 100 gradians, making certain calculations in surveying and civil engineering cleaner. To convert degrees to gradians, multiply by

\frac{10}{9}

. To convert gradians to radians, multiply by

\frac{\pi}{200}

. Most scientific calculators have a GRA or GRAD mode alongside DEG and RAD — if your trig answers look wrong, check which mode your calculator is set to.

Worked Example

Problem: Convert 45° to gradians.

Apply the conversion: Multiply the degree measure by 10/9.

G = 45 \times \frac{10}{9} = \frac{450}{9} = 50

Answer: 45° equals 50 gradians.

Why It Matters

Gradians appear most often in surveying, land measurement, and some European engineering contexts. On standardized tests and in trig courses, you need to recognize gradians so you can verify your calculator is set to the correct angle mode before computing sine, cosine, or tangent.

Common Mistakes

Mistake: Leaving a calculator in GRAD mode when a problem expects degrees or radians.

Correction: Always check your calculator's angle mode before evaluating trig functions. For example, sin(90) returns 1 in DEG mode but approximately 0.987 in GRAD mode.