gradians (grad) to radians (rad) conversion

gradians to radians conversion table

gradians (grad)radians (rad)
00
10.01570796326795
20.0314159265359
30.04712388980385
40.0628318530718
50.07853981633974
60.09424777960769
70.1099557428756
80.1256637061436
90.1413716694115
100.1570796326795
200.314159265359
300.4712388980385
400.628318530718
500.7853981633974
600.9424777960769
701.0995574287564
801.2566370614359
901.4137166941154
1001.5707963267949
100015.707963267949

How to convert gradians to radians?

Here's a breakdown of how to convert between gradians and radians, focusing on clarity, conciseness, and SEO best practices.

Understanding Gradians and Radians

Gradians (also called "grades" or "gons") and radians are both units used to measure angles. A gradian is defined as 1400\frac{1}{400} of a circle, while a radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. These units are part of the broader topic of angular measurement, crucial in fields like navigation, engineering, and physics.

Converting Gradians to Radians

To convert gradians to radians, you can use the following formula:

Radians=Gradians×π200\text{Radians} = \text{Gradians} \times \frac{\pi}{200}

So, to convert 1 gradian to radians:

Radians=1×π200=π2000.015708 radians\text{Radians} = 1 \times \frac{\pi}{200} = \frac{\pi}{200} \approx 0.015708 \text{ radians}

Converting Radians to Gradians

To convert radians to gradians, use the reciprocal of the above conversion factor:

Gradians=Radians×200π\text{Gradians} = \text{Radians} \times \frac{200}{\pi}

Therefore, to convert 1 radian to gradians:

Gradians=1×200π63.662 gradians\text{Gradians} = 1 \times \frac{200}{\pi} \approx 63.662 \text{ gradians}

Historical Context and Interesting Facts

  • The French Revolution: The gradian (or grade) was introduced during the French Revolution as part of the metric system, with the aim of simplifying calculations. The idea was to divide a right angle into 100 grades, making calculations more decimal-friendly.

  • Radian's Natural Significance: Radians are preferred in many mathematical and scientific contexts because they are directly linked to the circle's radius and circumference, simplifying many formulas in calculus, physics, and engineering.

Real-World Examples

While gradians aren't as widely used as degrees or radians, here are some examples where you might encounter conversions:

  • Surveying: Surveying equipment sometimes provides angles in gradians, requiring conversion to radians or degrees for calculations.
  • Some Engineering Applications: Certain specialized engineering tools might use gradians, necessitating conversion to radians for use with standard formulas.
  • Navigation Systems (Historically): Though less common now, some older navigation systems may have used gradians.

Example Scenario:

An engineer is using a surveying tool that gives an angle of 50 gradians. To perform calculations in a system that uses radians, they need to convert:

Radians=50×π200=π40.7854 radians\text{Radians} = 50 \times \frac{\pi}{200} = \frac{\pi}{4} \approx 0.7854 \text{ radians}

See below section for step by step unit conversion with formulas and explanations. Please refer to the table below for a list of all the radians to other unit conversions.

What is gradians?

Gradians, also known as gons, are a unit of angular measurement primarily used in surveying, civil engineering, and some European countries. This section explores the definition, formation, and applications of gradians.

Definition of Gradians

A gradian is defined as 1400\frac{1}{400} of a full circle. This means there are 400 gradians in a complete rotation. It's an alternative to degrees (360 in a full circle) and radians (2π2\pi in a full circle). The symbol for gradian is "gon" or "grad".

Formation and Relationship to Other Angle Units

The gradian system was introduced in France around the time of the French Revolution as part of the metric system, aiming for a decimal-based approach to angle measurement.

  • Relationship to Degrees: 1 full circle = 360 degrees = 400 gradians. Therefore, 1 gradian = 360400\frac{360}{400} = 0.9 degrees.
  • Relationship to Radians: Since 2π2\pi radians = 400 gradians, 1 gradian = 2π400\frac{2\pi}{400} = π200\frac{\pi}{200} radians.

The appeal of gradians lies in their decimal-friendly nature. A right angle is exactly 100 gradians, which can simplify calculations in certain contexts.

Historical Context and Notable Figures

While the gradian system was intended to integrate seamlessly with the metric system, it didn't achieve widespread adoption globally. While no single individual is directly credited with "discovering" or "inventing" the gradian in the same way someone might discover a physical law, its creation is associated with the general movement towards decimalization that occurred during the French Revolution. The French committee that developed the metric system advocated for its use.

Real-World Examples and Applications

  • Surveying: Surveying equipment, particularly in Europe, often provides angle readings in gradians. This can simplify calculations when dealing with slopes and distances. For example, a slope of 1 gradian represents a rise of 1 meter for every 100 meters of horizontal distance.
  • Civil Engineering: Similar to surveying, civil engineering projects may utilize gradians for calculations related to land gradients and construction angles.
  • Navigation and Mapping: While less common, some navigation systems and mapping software may offer the option to display angles in gradians.

Conversion Formulas

  • Gradians to Degrees:

    Degrees=Gradians910Degrees = Gradians * \frac{9}{10}

  • Degrees to Gradians:

    Gradians=Degrees109Gradians = Degrees * \frac{10}{9}

  • Gradians to Radians:

    Radians=Gradiansπ200Radians = Gradians * \frac{\pi}{200}

  • Radians to Gradians:

    Gradians=Radians200πGradians = Radians * \frac{200}{\pi}

What is radians?

Radians are a fundamental unit for measuring angles, particularly useful in mathematics and physics. They provide a natural way to relate angles to the radius of a circle and are essential for various calculations involving circular motion, trigonometry, and calculus.

Understanding Radians

A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. In simpler terms, imagine taking the radius of a circle and bending it along the circumference. The angle formed at the center of the circle by this arc is one radian.

Radian Formation

To visualize how radians are formed:

  1. Start with a circle: Draw any circle with a defined radius, rr.
  2. Measure the radius along the circumference: Take the length of the radius, rr, and mark off that same length along the circumference of the circle.
  3. Draw the angle: Draw two lines from the center of the circle to the start and end points of the arc you just marked.
  4. The angle is one radian: The angle formed at the center of the circle is one radian.

Since the circumference of a circle is 2πr2\pi r, there are 2π2\pi radians in a full circle (360360^\circ). Therefore:

2π radians=3602\pi \text{ radians} = 360^\circ

1 radian=3602π57.29581 \text{ radian} = \frac{360^\circ}{2\pi} \approx 57.2958^\circ

Conversions between Radians and Degrees

  • Degrees to Radians: To convert an angle from degrees to radians, multiply the angle in degrees by π180\frac{\pi}{180}:

    Radians=Degrees×π180\text{Radians} = \text{Degrees} \times \frac{\pi}{180}

  • Radians to Degrees: To convert an angle from radians to degrees, multiply the angle in radians by 180π\frac{180}{\pi}:

    Degrees=Radians×180π\text{Degrees} = \text{Radians} \times \frac{180}{\pi}

Radian and Arc Length

One of the most important applications of radians is in calculating arc length. The arc length ss of a circle is given by:

s=rθs = r\theta

Where:

  • ss is the arc length
  • rr is the radius of the circle
  • θ\theta is the angle in radians

Interesting Facts and Laws

  • Simplicity in Calculus: Radians simplify many formulas in calculus, especially those involving trigonometric functions. For example, the derivative of sin(x)\sin(x) is cos(x)\cos(x) only when xx is measured in radians.
  • Natural Unit: Radians are considered a "natural" unit for measuring angles because they directly relate the angle to the properties of a circle.
  • Euler's Identity: One of the most famous equations in mathematics, Euler's Identity, involves radians: eiπ+1=0e^{i\pi} + 1 = 0. This equation connects five fundamental mathematical constants.

Real-World Applications

  • Circular Motion: In physics, radians are used to describe angular velocity (ω\omega) and angular acceleration (α\alpha) in circular motion. For instance, angular velocity is often measured in radians per second (rad/s).

    ω=ΔθΔt\omega = \frac{\Delta\theta}{\Delta t}

  • Navigation: Radians are used in navigation systems, particularly in calculations involving latitude and longitude on the Earth's surface.

  • Signal Processing: Radians are used in Fourier analysis and signal processing to represent frequencies and phases of signals.

  • Computer Graphics: Radians are used extensively in computer graphics to specify rotations and angles for objects and transformations.

  • Pendulums: In the analysis of simple harmonic motion, such as a pendulum's swing, the angle of displacement is often expressed in radians for simpler mathematical treatment. For small angles, sin(θ)θ\sin(\theta) \approx \theta when θ\theta is in radians.

Complete gradians conversion table

Enter # of gradians
Convert 1 grad to other unitsResult
gradians to radians (grad to rad)0.01570796326795
gradians to degrees (grad to deg)0.9
gradians to arcminutes (grad to arcmin)54
gradians to arcseconds (grad to arcsec)3240