Rule

Relation between Radians and Degrees

Since both degrees and radians are used to measure angles, it's useful to be able to translate between them.

18 0^{\circ} = π rad

The arc length for one revolution around a circle is also known as the circumference of the circle,

2 π r .

By dividing this with the length of the arc corresponding to one radian,

r,

the number of radians for one revolution is obtained.

\frac{2 π r}{r} = 2 π

Thus, one revolution is

2 π

radians. In degrees, this is

36 0^{\circ},

leading to the relation

36 0^{\circ} = 2 π

rad. Dividing both sides by

2

gives

18 0^{\circ} = π rad .

Thus,

18 0^{\circ}

correspond to

π

radians.

Rule

Relations

From the relation

18 0^{\circ} = π

rad it's possible to find two rules, by dividing both sides by either

180

π .

1^{\circ} = \frac{π}{1 8 0} rad

To find an expression for $1^{\circ},$ divide both sides by $180 .$

18 0^{\circ} = π rad

$LHS / 180 = RHS / 180$

\frac{1 8 0}{1 8 0}^{\circ} = \frac{π}{1 8 0} rad

Calculate quotient

1^{\circ} = \frac{π}{1 8 0} rad

1^{\circ}

corresponds

\frac{π}{1 8 0} \approx 0.017

radians. That means that

2^{\circ} = 2 \cdot \frac{π}{1 8 0} rad 3^{\circ} = 3 \cdot \frac{π}{1 8 0} rad,

and so forth.

1 rad = \frac{1 8 0 ^{\circ}}{π}

To get an expression for $1$ rad, divide both sides by $π .$

18 0^{\circ} = π rad

$LHS / π = RHS / π$

\frac{1 8 0 ^{\circ}}{π} = \frac{π}{π} rad

Calculate quotient

\frac{1 8 0 ^{\circ}}{π} = 1 rad

Rearrange equation

1 rad = \frac{1 8 0 ^{\circ}}{π}

$1$ radian corresponds to $\frac{1 8 0}{π} \approx 57 . 3^{\circ} .$

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