Angle Conversion

An Angle is formed when two lines, called rays, meet at a common point known as the vertex. The amount of "turn" between the two rays is what we measure when we measure an angle.

There are several ways to measure angles, using different units:

1. Degrees (°):

   • The most commonly used unit for measuring angles.
   • A full circle is divided into 360 equal parts, so one full revolution equals 360 degrees.
   • Key Angles:
     • 90°: A right angle, like the corner of a square.
     • 180°: A straight angle, forming a straight line.
     • 360°: A full angle or full rotation.
   • Degrees are used in everyday situations, like in geometry, navigation, and telling time.

2. Radians (rad):

   • The standard unit of angle measurement in mathematics, especially in calculus and trigonometry.
   • One radian is the angle formed when the length of the arc is equal to the radius of the circle.
   • A full circle is 2π radians, so:
     • 90° = π/2 radians.
     • 180° = π radians.
     • 360° = 2π radians.
   • Radians are useful because they naturally relate angles to the properties of circles.

3. Gradians (gon or grad):

   • A less common unit of angle measurement.
   • A full circle is divided into 400 gradians, so:
     • 90° = 100 gradians.
     • 180° = 200 gradians.
     • 360° = 400 gradians.
   • Gradians are sometimes used in surveying and civil engineering because they divide a right angle into an even 100 parts.

4. Revolutions (rev):

   • A full turn or rotation around a point is measured as one revolution.
   • One full revolution is equivalent to:
     • 360° (degrees),
     • 2π radians,
     • 400 gradians.
   • Revolutions are commonly used in contexts like gears, wheels, and rotational motion.

Imagine standing in the center of a large circle. If you turn completely around, you've made one full revolution:

• In degrees, that’s 360°.
• In radians, that’s 2π radians.
• In gradians, that’s 400 gradians.
• In revolutions, that’s 1 revolution.

Understanding the relationships between these units allows you to convert from one to another:

• Degrees to Radians: Multiply by π/180.
• Radians to Degrees: Multiply by 180/π.
• Degrees to Gradians: Multiply by 10/9.
• Gradians to Degrees: Multiply by 9/10.

• Protractor: Typically marked in degrees, used to measure angles on paper.
• Compass and Ruler: Used to draw angles and circles.
• Scientific Calculator: Often used for converting between different units like degrees and radians.

• Degrees (°): Most common, everyday use.
• Radians (rad): Preferred in higher mathematics.
• Gradians (gon): Specialized uses like surveying.
• Revolutions (rev): Describes full rotations.

Understanding these different units and how they relate to each other is crucial in fields ranging from basic geometry to advanced engineering and physics.