Extension of measurement of angles

Positive and negative angles

We have already introduced positive and negative angles into geometry. If the hand of a clock is to be turned by a certain angle, you must still tell whether it is to be turned forwards or backwards. In a similar manner as it was done with positive and negative numbers, you can also agree to talk about forward and backward or negative and positive angles. The definition is unimportant, since, in making this definition, it is only important to define a definite rule for the leading signs, and not to change it in the course of an investigation.

A rotation which is opposite to that of the clock is, as a rule,
called positive, that with the clock negative

The starting and finishing legs of angles must be distinguished. You can also turn a clock's hand by more than 360º, so that you can now think about arbitrarily large and small positive and negative angles. Instead of +a we will now simply write a.

Definition: We will understand by the arc measure of an angle or arc a the length of the corresponding arc of the unit circle (Latin: arcus)

In the figure shown here, arc a is the length of b. Since the circumference of a circle is r and that of the unit circle , you have

angle		measure		arc
360º				arc 360º =
180º				arc 180º =
90º		/2		arc 90º = /2

a and b grow uniformly, whence during calculations by proportions or by the functional method the measure of an angle and its arc are readily converted into each other. You can also employ proportions freely.

a^º/180º = arc a/p.

arc a and p are pure numbers, a and the stretched angle must have the same unit, the same measure of the angle, for example, degrees.

Task: Find arc 60º.

If a is measured in degrees, then quite generally: arc a = p ·a/180

Task: Find a in degrees for arc a = 1.

a/180 = 1/p

a = 180/p = 57.296

a 57.3º

If a is measured in degrees, then, in general, a = arc a ·180/p.

We have chosen the radius as a unit of measure, whence the arc measure of an angle states how often the measure unit or the radius is contained in the corresponding arc. If you select another unit, you must give the definition the form: arc a= b/r.

There will be no contradictions, because all circles are similar to each other, whence equally located items are in the same ratio, that is b₁/r₁ = b₂/r₂ = b₃/r₃ = ···,

Task: Compute the length of the arc, which corresponds to the central angle arc a= 2.4, if the radius of the circle is r = 10 cm.

Denoting the measure of the arc of a by x, you can rewrite the formulae

or

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