Convert Angle from radians to gradians (rad to grad)

radian to gradian Conversion Formula

Example

Let's convert 5 radian (rad) to gradian (grad).

Using the formula:

Therefore, 5 radian (rad) is equal to 318.3098861837906957 gradian (grad).

How many radians are in one gradian? One gradian (grad) contains 0.015707963267948965 radians (rad) — the inverse of the factor above. Multiplying by 63.66197723675813914 takes you from radians to gradians; multiplying by 0.015707963267948965 brings you back.

Put in words: one radian equals 63.66197723675813914 gradians, so the radian is the larger unit of this pair. Converting between them never changes the amount of angle being measured — only the size of the unit you count it in.

What is a radian (rad)?

A radian (rad) is the standard unit used to measure angles, and it's part of the International System of Units (SI).

One radian is equal to roughly 57.3 degrees (deg), or more precisely, 180/π degrees.

Why Do We Use Radians?

The radian is the official SI unit for angles, making it the preferred choice in science and engineering.

It is especially common in fields like physics, calculus, and computer graphics. The main advantages of using radians are:

• They simplify many essential mathematical and scientific formulas.
• They create a direct link between a circle's angle and its arc length and radius.
• They are the standard for higher-level mathematics, ensuring consistency in calculations.

How to Convert Radians to Degrees

A full circle has 360 degrees (360°), which is equal to 2π radians.

This key relationship gives us the precise formulas for converting between radians and degrees:

• To convert radians to degrees: Degrees = Radians × (180/π)
• To convert degrees to radians: Radians = Degrees × (π/180)

Here are some common angle conversions:

What is the Definition of a Radian?

Geometrically, the definition of a radian is based on a circle's own properties.

Imagine an arc along the edge of a circle that has the same length as the circle's radius. The angle created at the center of the circle by this specific arc is exactly one radian.

Because a radian is a ratio of two lengths (arc length divided by radius), it is technically a dimensionless unit. This unique property is why it integrates so smoothly into advanced math and physics formulas.

What is a Gradian?

While most of us learn about measuring angles in degrees, there's another unit of angle measurement called the gradian.

A gradian (often abbreviated as grad) is a unit of angle equal to 0.9 (or 9/10) of a degree. It's an alternative to degrees and radians, designed to make some calculations simpler.

How Many Gradians Are in a Circle?

Unlike the familiar 360-degree system, a full circle is divided into 400 gradians.

This base-100 system simplifies many geometric calculations:

• A right angle is exactly 100 gradians.
• A straight angle is 200 gradians.
• A three-quarter turn is 300 gradians.

This makes it an intuitive framework for certain types of angular math.

How Do You Convert Gradians to Degrees?

The conversion between gradians and degrees is straightforward.

Here is the simple gradians to degrees formula:

• Gradians to Degrees: Multiply the gradian value by 0.9.
• Degrees to Gradians: Multiply the degree value by 10/9 (approximately 1.11).

For example, to convert 200 gradians to degrees, you would calculate: 200 grad × 0.9 = 180 deg.

What is the Relationship Between Gradians and Radians?

When comparing radians vs. gradians, the relationship is key for trigonometry and higher math.

A full circle is 400 gradians or 2π radians.

This means that 200 gradians are equal to π radians. The conversion factor, π/200, is fundamental for calculations that bridge these two systems.

Why Use Gradians?

So, why use this system at all?

Gradians are primarily used in specific professional fields like surveying and some branches of engineering, especially in Europe.

The decimal-based nature of the 400-gradian circle simplifies calculations involving right angles, making fieldwork and topographical analysis more efficient.

Where Gradians Survive Today

The gradian is the rarest of the three common angle units, but it has not disappeared — it has retreated into a few specific places.

• Continental European surveying. French, Swiss, Austrian and some German cadastral practice still works in gradians, where the unit is often called the gon. A quadrant is exactly 100 gon, so a bearing can be read and written without ever leaving decimal arithmetic.
• Total stations and survey instruments. Many instruments ship with a unit selector offering degrees, gradians and mils, precisely because a crew may have to match whatever the existing site records use.
• Calculators. The DEG / RAD / GRAD switch on nearly every scientific calculator is the unit's most widely seen trace — and the most common way people encounter it by accident.

That last point is worth a warning. A calculator left in GRAD mode returns answers that look reasonable but are wrong by a factor of 0.9 against degrees. If a trigonometric result is off by roughly ten percent and nothing else explains it, check the mode before checking the formula.

The quick conversions are:

• 1 gradian = 0.9 degrees; 1 degree = 1.111… gradians
• A right angle is 100 gradians, a full turn is 400
• 1 gradian ≈ 0.015708 radians

radian to gradian Conversion Table

Here are some quick reference conversions from radian (rad) to gradian (grad):