Question

What is the basic unit by which angles are measured?

Answer 1

There are two units of angle measurement - degrees (`360^o` in a full angle) and radians (`2pi` radians in a full angle).

Explanation:

Imagine a circle and a fixed radius in it. Start rotating this radius from its initial position counterclockwise. This is called the positive direction of a change of an angle from the initial position of this radius to its current position. The measurement of this angle is a positive number.

Opposite direction (clockwise) is called the negative direction and angles from the initial position of a radius to its current position are measured as negative numbers.

When our radius reaches its initial position, thus making one full rotation, the angle between the final position and the initial one is called a full angle.

One unit of an angle measurement is called a degree (written as `1^o`) and it's defined as `1/360` of a full angle.
Half of the full angle, called straight angle, when initial and final position of our radius are on the same line, but directed to opposite direction, is, therefore, measured as `180^o`.
A quoter of a full angle, called right angle, when initial and final positions of our radius are perpendicular to each other, is measured as `90^o`.

Another unit of measurement is called radian and it's defined as `1/(2pi)` of a full angle. It is convenient because, if a radius `R` of a circle is given and an angle `phi` between two radii to two endpoints of an arc is in radians, the length of an arc would constitute `phi/(2pi)` part of a total circumference of a circle and will be equal to
`(phi/(2pi))*2piR = R*phi` - a short and nice formula.
There are some other (and more important) reasons for choosing this unit of measurement.

Half of the full angle, called straight angle, when initial and final position of our radius are on the same line, but directed to opposite direction, is, therefore, measured as `pi` radians.
A quoter of a full angle, called right angle, when initial and final positions of our radius are perpendicular to each other, is measured as `pi/2` radians.

If an angle is specified as a real number without a sign of degree near it (like `25`, not `25^o`), it is assumed to be in radians.