Angle Conversion Calculator
A comprehensive tool to convert angles between various units, including degrees, radians, gradians, and revolutions.
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Understanding and Converting Angles
A guide to the most common units of angle measurement—Degrees, Radians, and Gradians—and how to convert between them.
The Units of Angle Measurement
Angles are a fundamental concept in geometry, trigonometry, physics, and engineering. While we often think of angles in degrees, there are several other units used in different contexts. This calculator helps convert between the three most common ones.
- Degrees (°): The most familiar unit. A full circle is divided into 360 degrees. This system dates back to ancient Babylon.
- Radians (rad): The standard unit of angular measure in mathematics and physics. A radian is defined by the arc of a circle: one radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius. A full circle is 2π radians.
- Gradians (grad): Also known as "gon" or "grade," this unit divides a right angle into 100 parts. A full circle is 400 gradians. It is sometimes used in surveying and construction.
Conversion Formulas
All conversions are based on the relationship of each unit to a full circle:
360° = 2π radians = 400 gradians
From this, we can derive the key conversion factors:
- Degrees to Radians: Radians = Degrees × (π / 180)
- Radians to Degrees: Degrees = Radians × (180 / π)
- Degrees to Gradians: Gradians = Degrees × (400 / 360) or Degrees × (10 / 9)
- Gradians to Degrees: Degrees = Gradians × (360 / 400) or Gradians × (9 / 10)
- Radians to Gradians: Gradians = Radians × (200 / π)
Frequently Asked Questions (FAQ)
- Q: Why use radians instead of degrees?
- A: Radians are the "natural" unit for angles. Many formulas in calculus and physics, especially those involving trigonometric functions and rotational motion, are simpler and more elegant when expressed in radians. For example, the derivative of sin(x) is cos(x) only when x is in radians.
- Q: What is π (pi)?
- A: Pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.14159.
- Q: Where are gradians used?
- A: Gradians are primarily used in some fields of surveying and civil engineering. Because a right angle is exactly 100 gradians, it can simplify some calculations involving perpendicular lines.
- Q: How many degrees are in one radian?
- A: One radian is approximately 57.3 degrees (180/π).
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