Radians - Circular measure of Angle

The radian is the angle seen from the arc of the circle

The radian is the angle seen from the arc of the circle

For example, put the radius r = 1 on the circumference as it is.
The angle at that time is 1 [rad].

Angle 360 degrees

Degree measure, which divides the circle into 360 degrees, is intuitive.

Essence of radian

r = 1

Let's check with a circle with r = 1

When the arc length is 1, the angle is 1 [rad].

When the arc length is 2, the angle is 2 [rad].
When the angle is 2 [rad], the length of the arc is 2.

When the arc length is 3, the angle is 3 [rad].
When the angle is 3 [rad], the length of the arc is 3.

When the arc length is 3.1415 ..., it represents a semicircle.
The angle that represents a semicircle in radians is π [rad]
When the angle is π[rad] (3.1415..), the length of the arc is π.
π is the circular constant.

When the arc length is 2π, the angle is 2π [rad].
When 2π [rad], the arc is a perfect circle and the length is 2π.

Beautiful Radians

Q : Why use radians?

A : Because it's easy

You can easily express the characteristics of a circle without using the angle method 360 or pi.

Q : For example?

A : Arc length

Let ' l ' be the length of the arc,

Let Θ be a radian,

l = r Θ

Let Θ be a degree,

l = (2 π / 360) r Θ

A: Fan-shaped Area

Let Θ be a radian,

S = 1/2 r^2 Θ

Let Θ be a degree,

S = π / 360 r^2 Θ

Relationship between 360 and rad

Angles can be expressed in either radians or degrees.
However, the viewpoint is different.

The angle of a perfect circle is the same in both the radian method and the radian method.
Radian 2π [rad]
Degree 360°

2π [rad] ＝ 360°

1 [rad] = 180° / π

Extra! : Radius

Radian means "angle to make an arc" between two radii.

There is an opinion that 'Radian' was introduced by the engineer James Thomson in the 19th century. In fact, it seems that it was already used by Roger Cotes, who interacted with Newton in the 17th and 18th centuries.

Roger Cotes - Wikipedia

radius, radial, radii, radian, etc... everyone has something to do with the radius. Sometimes the radius is hidden in the words that are caught in the ear. For example, radial tires, radial loads, radial designs, radial antennas, radial axial, and so on.

For example, Radial Tire has a structure in which the tire frame is arranged in the radial direction!

Summary

- Radian represents the angle to the arc
- The angle of the radian method corresponds to the arc of a circle
- Placing the radius of a circle with a radius of 1 on the arc of the circle makes it intuitively easy to understand.
- Using radians, you can simply express the characteristics of a circle.