What is Degree and Radians - It's Introduction, Examples, Formulas

Degree:

Degrees measure angles, angles measure how much something turns or rotates, like the slice of a pizza or the direction a wheel spins. Degrees are an arbitrary, human invented unit for measuring angles. We arbitrarily decided a circle should have 360 parts because it has many divisors. The old ancient civilizations like the Babylonians liked numbers that divide nicely i.e. 360 has many factors: 1, 2, 3, 4, 5, 6, etc.. It might relate to the ~365 days in a year.

You should use degrees when you are measuring angles using a protractor, or describing a physical picture. Most people have developed intuitive feel for the common angles. This would be common in vector related problems, including speeds, projectiles, forces, and similar situations. Best for basic geometry, directions, or when you want simple numbers. e.g. clocks: 360°/12 hours = 30° per hour. Degrees are for quick, practical use. Use degrees for everyday life and practical measurements.

e.g.

1. Imagine a circle as a pizza degrees are like cutting it into slice. Think of a the whole pizza as being divided into 360 equal parts. Each part is 1⁰. A straight line or half circle is 180°. A right angle is 90° i.e. one quarter of the circle. Turning all the way around is 360°. Degrees based on whole numbers and common fractions, they feel human friendly because.
2. Imagine a bicycle wheel. You look at the wheel and say, If I turn the wheel one degree, I'm turning it 1/360th of a full rotation. It doesn't tell you how far the bike will move.

Radians:

Radians actually measure distance. Radians are a natural, mathematical unit for measuring angles based on the geometry of the circle itself. A radian measures an angle by the distance traveled around the circle. This makes the radian particularly useful when talking about rotational motion. For example, saying that you moved a distance of 2 radians makes a lot more sense than saying you moved a distance of, let's say, 40°.

• Radians used to simplify formulas in calculus, physics, and engineering. For instance, in speed calculations like angular velocity, radians make the math cleaner without extra conversion factors.
• Radians used in advanced math/science where circles and waves are involved. e.g. sine waves in physics use radians naturally, avoiding clunky constants. Radians make equations elegant. e.g. the derivative of sin(x) is cos(x) only in radians.
• Use radians whenever you are doing math, physics, or anything involving calculus. They are the native language of angles for mathematics.
• Radians are most commonly used for descibing trigonometric functions.
Degree describes the angle between a light ray hitting a mirror and the surface would usually be given in degrees.
• You should use radians when you are looking at objects moving in circular paths or parts of circular path.  In particular, rotational motion equations are almost always expressed using radians.

e.g.

1. Imagine a circle as a pizza, radians are like measuring slices based on the pizza's own radius. Instead of cutting slices, radians tie the angle directly to the circle's geometry. 1 radian is the angle where the arc length equals the radius. Take a string as long as the radius and lay it along the curve of the circle that bend creates exactly 1 radian of angle at the center. A full circle's circumference is 2π times the radius, so the total angle is 2π radians i.e. about 6.28 radians. A straight line is π radians i.e. 3.14 radians. A right angle is π/2 radians i.e. 1.57 radians.

2. Imagine a bicycle wheel. You look at the wheel and say, If I use the length of the wheel's spoke as a measuring stick, how many spoke-lengths did the wheel roll? Take the radius of a circle (r). Wrap that radius length around the circle's circumference. The angle that cuts off an arc exactly one radius long is 1 radian. The circumference of a circle is 2πr. So, you can fit 2π i.e. approximately 6.28 radian, radius lengths around the circumference. Therefore, a full circle, which is 360°, is equal to 2π radians.

If you walk along the edge of a circle with steps equal to the radius, you'll take about 6.28 steps to go full circle, that's 2π radians.

Conversion:
The key conversion is:
180° = π radians

• Convert degrees to radians: Multiply by π/180
e.g. 90° * (π/180) = π/2 radians
• Convert radians to degrees: Multiply by 180/π
e.g. π/4 radians * (180/π) = 45°