An angle is the “opening” between two rays of the same origin. To measure it, we transport the question to an arc: the angle measures the same as the arc it determines in a circle centered at its vertex.

The story

The The Elements of Euclid (c. 300 BC) defines the plane angle as “the inclination, relative to each other, of two lines in a plane that meet and are not in a straight line”. The definition seems simple, but the Greeks discussed for centuries what is, after all, an angle: a quantity, which can be added and compared? a quality, like a form? or a relation between two lines? The person who tells this dispute is Proclus, in the 5th century, in his commentary on Euclid. The solution we adopted — measuring the angle by the arc it determines in a circle centered on the vertex — leaves the philosophical question aside and transforms the angle into a number.

The clock face also has ancient roots: the division of day and night into 12 hours comes from the Egyptians, and the 60 minutes of each hour from the base 60 of the Babylonians. The minute hand, however, only became common after Christiaan Huygens built the pendulum clock, in 1656, the first accurate enough for it to make sense.

Figure from the *School of Athens*, by Raphael, traditionally identified as Eudemus of Rhodes; uncertain identification.
Figure from the School of Athens, by Raphael, traditionally identified as Eudemus of Rhodes; uncertain identification.Rafael Sanzio; clipping of reproduction by Jic and FranksValli · Wikimedia Commons · Public domain
How Thales calculated the distance of a ship, according to Eudemus: from the top of a tower $H$ high, one looks at the ship; triangles $TKJ$ and $TBN$ are similar, then $D/H = d/h$.

The notation of angles. Euclid wrote “the angle $ABC$” in full. Signs for “angle” and “right angle” only appeared in the 17th century, and for more than two centuries each author used his own:

  1. 1618
    Kepler, in Epitome astronomiae Copernicanae, uses a sign for the right angle.
  2. 1634
    Pierre Hérigone, in Cursus mathematicus, creates signs for “angle” and “right angle” — his book is full of new symbols.
  3. 1672–1698
    William Leybourn, John Kersey and Samuel Reyher propose other forms in their geometry and algebra books.
  4. 1794
    Legendre, in Éléments de géométrie, one of the most read geometry books of the 19th century.
  5. 1827
    Möbius, in Der barycentrische Calcul.
  6. 1847
    Oliver Byrne publishes Euclid’s The Elements in color, with the angles drawn rather than named.
  7. 1897
    Giuseppe Peano, in Formulario mathematico, attempts to standardize all mathematical notation.

Sources: Eudemus (MacTutor) · Euclid (MacTutor) · Aulus Gélio, Attic Nights, XIII.5 · Euclid, The Elements, book I, definition 8 · F. Cajori, A History of Mathematical Notations, vol. I (1928).

In everyday life

  • Maneuvers. In skateboarding and snowboarding, maneuvers are named after angles: a 180 is a half-turn, a 360 is a full turn and a 540 is a turn and a half.
  • Airport runways. The number painted on the threshold is the runway heading divided by 10: the runway 09 points to $90^\circ$ (east) and, seen from the other side, is 27, heading $270^\circ$ (west).
  • Television. It is recommended that the screen occupies between $30^\circ$ and $40^\circ$ of the field of view; Therefore, the ideal distance to the sofa depends on the size of the TV.
Bearings measured in degrees: move the bearing and see where the components point. Airport runways and maps use the same idea.

Theory

Angles of two rays

Consider two rays $Oa$ and $Ob$ with the same origin $O$, distinct and not opposite.

  • The straight line that contains $Oa$ divides the plane into two opposite half-planes: $\alpha$, which contains $Ob$, and $\alpha'$, which does not contain $Ob$.
  • The line that contains $Ob$ divides the plane into two opposite half-planes: $\beta$, which contains $Oa$, and $\beta'$, which does not contain $Oa$.

DefiniçãoConvex angle and concave angle

  • The convex angle $a\hat{O}b$ is the intersection of the half-planes $\alpha$ and $\beta$: $\;a\hat{O}b = \alpha \cap \beta$.
  • The concave angle $a\hat{O}b$ is the meeting of the half-planes $\alpha'$ and $\beta'$: $\;a\hat{O}b = \alpha' \cup \beta'$.
The convex angle (on the left) is the common part of $\alpha$ and $\beta$; the concave angle (on the right) is the meeting of $\alpha'$ and $\beta'$ — everything outside the convex, plus the sides.

DefiniçãoParticular cases

  • If the rays $Oa$ and $Ob$ coincide, they determine two angles: the null angle and the one-turn angle.
  • If the rays $Oa$ and $Ob$ are opposite, they determine two shallow angles.

Measure of angles

Given an angle $a\hat{O}b$, take a circle with center $O$ and radius $r$, and let $A$ and $B$ be the points where the sides of the angle intersect it. Each arc $\overset{\frown}{AB}$ corresponds to a single central angle $a\hat{O}b$, and vice versa.

DefiniçãoMeasure of an angle

We agree that a unitary arc corresponds to a unitary central angle. So the arc $\overset{\frown}{AB}$ and the central angle $a\hat{O}b$ have the same measure.

  1. The angle of $1^\circ$ is the central angle that determines an arc equal to $\dfrac{1}{360}$ on the circle.
  2. The 1 rad angle is the central angle that determines an arc of length equal to the radius.
  3. The angle of $60^\circ$ is the central angle corresponding to an arc of $60^\circ$.
  4. The $\pi$ rad angle is the central angle corresponding to an arc of $\pi$ rad (the shallow angle).

To measure an angle $a\hat{O}b$ in radians, we construct a circle with center $O$ and radius $r$ and check how many radians the arc $\overset{\frown}{AB}$ measures, that is, we divide the length $\ell$ of the arc by the radius:

PropriedadeAngle in radians

$$\alpha = \frac{\ell}{r} \qquad (\alpha \text{ em radianos}).$$

Equivalently, the length of an arc of central angle $\alpha$ rad is $\ell = \alpha \cdot r$.

ExemploAngle of an arc of 8 cm

If the central angle $a\hat{O}b$ determines, in a circle of radius $r = 5$ cm, an arc of length $\ell = 8$ cm, then

$$\alpha = \frac{\ell}{r} = \frac{8}{5} = 1{,}6 \text{ rad}.$$

The measurement does not depend on the chosen circumference. Once the angle $a\hat{O}b$ of measure $\alpha$ rad is fixed and circles with centers $O$ and radii $r_1, r_2, r_3, \ldots$ are constructed, the corresponding arcs have lengths $\ell_1, \ell_2, \ell_3, \ldots$ such that

$$\frac{\ell_1}{r_1} = \frac{\ell_2}{r_2} = \frac{\ell_3}{r_3} = \cdots = \alpha.$$
Drag the yellow dot to open or close the angle. The three arcs have different lengths, but the ratio between arc and radius is the same: the measure of the angle in radians.
Why doesn't the ratio depend on the radius?

Two circles with the same center are similar figures: one is an enlargement of the other, with ratio $k = \dfrac{r_2}{r_1}$. In an enlargement, all lengths are multiplied by $k$, including the arc determined by the same angle: $\ell_2 = k\,\ell_1$. Logo

$$\frac{\ell_2}{r_2} = \frac{k\,\ell_1}{k\,r_1} = \frac{\ell_1}{r_1}.$$

Angles on the clock

A clock face is a circle divided into 12 equal parts of $30^\circ$. The large hand moves through $360^\circ$ in 60 minutes ($6^\circ$ per minute); the small one travels $30^\circ$ in 60 minutes, that is, $0{,}5^\circ$ per minute. These two facts solve the angle problems between pointers.

Choose the hour and minutes. The golden arc is the smallest angle between the hands.

The exercises in this section are numbers 11 to 16 from chapter list.