Multiples and Submultiples of an Angle

A multiple of an angle α is an angle β that is congruent to α when multiplied by n. $$ \beta = \alpha \cdot n $$

Here, n is a natural number or, alternatively, a rational number expressed as n=m/q.

For example, let's consider an angle alpha of 15°.

$$ \alpha = 15° $$

The angle beta, which is 45°, is a multiple of alpha because it is congruent to alpha multiplied by n=3.

$$ \beta = 45° = \alpha \cdot 3 = 15° \cdot 3 $$

A submultiple of an angle β is an angle α that is congruent to the n-th part of β. $$ \alpha = \frac{1}{n} \cdot \beta $$

In the previous example, the angle α=15° is a submultiple of the angle β=45° because it is congruent to one-third of beta.

$$ \alpha = 15° = \frac{1}{3} \cdot \beta = \frac{1}{3} \cdot 45° $$

And so forth.