Angle Measure: Degrees and Radians

The Radian: When the Arc Is as Long as the Radius

Imagine you take the radius of a circle and bend it along the edge, like a piece of string. The central angle that this arc, exactly as long as the radius, subtends is one radian. It doesn't matter whether the circle is small or huge: the ratio between arc and radius is the same for a given angle.

The full circumference measures $2\pi r$ and each radian covers an arc of length $r$; that is why exactly $2\pi$ radians fit in one full turn. So $2\pi$ rad $=360^\circ$ is no coincidence: it comes from the definition.

Converting Between Degrees and Radians

One full turn is $360^\circ$ and also $2\pi$ rad. That equivalence is the proportion that converts in either direction.

Degrees, minutes, and seconds

When a result in degrees doesn't come out exact, it is often expressed in degrees, minutes, and seconds: $1^\circ=60'$ (minutes) and $1'=60''$ (seconds). It works just like hours: the decimal part is multiplied by $60$ to get the next smaller unit.

For exercise MEDIO-02, first calculate the angle in radians, convert it to decimal degrees, and then split it into degrees, minutes, and seconds as above.

The regular octagon in exercise 07 uses this: the sum of the interior angles of a polygon with $n$ sides is $(n-2)\cdot180^\circ$, and in a regular polygon it is split into equal parts.

Full Turns, Negative Angles, and the Quadrant

An angle can be larger than one full turn or negative (a turn in the clockwise direction). To find where it ends, add or subtract full turns ($360^\circ$ or $2\pi$ rad) until it lies between $0$ and one full turn.

To locate the quadrant: $0$ to $90^\circ$ is I, $90^\circ$ to $180^\circ$ is II, $180^\circ$ to $270^\circ$ is III, and $270^\circ$ to $360^\circ$ is IV.

Arc Length and Area of a Circular Sector

Here the radian shows its usefulness: since $1$ rad is an arc equal to the radius, an arc of $\theta$ radians measures $\theta$ times the radius.

Angular Velocity: How Many Radians per Second

When something rotates (a disc, a wheel, the beam of a lighthouse), what matters is how much angle it sweeps per unit of time: this is the angular velocity, measured in rad/s.

This same reasoning works for pulleys connected by a belt: the points on the rims of both pulleys travel the same arc $s$, so $r_1\theta_1=r_2\theta_2$.

Back to the beginning: if a Ferris wheel turns through an angle $\theta$ in radians, a cabin at distance $r$ from the axis travels an arc $s=r\theta$.

The Hands of a Clock

An analog clock has two "discs" turning at different angular velocities. The minute hand makes one turn ($2\pi$ rad) in $60$ min; the hour hand makes one turn in $12$ h, that is, in $720$ min.

Shrinking Arcs: Summing an Infinite Number of Arcs

A pendulum with friction doesn't swing the same way forever: each swing is a little shorter than the previous one. To find how far it travels in total, you have to add up many arcs, each one a fixed fraction of the previous one.

Look at a simple case: $100+50+25+12.5+\dots$, where each term is half the previous one. The terms get so small that the sum doesn't grow without bound, but approaches a value. Which one? Call the total sum $S$. If you remove the first term, what remains is exactly half of $S$ (each term is half the previous one). Then:

The key idea: what remains after the first term is the same sum, multiplied by the fixed fraction $r$. That is, $S=a+rS$. This works when $r$ is between $0$ and $1$, because that is when the terms really do get shorter.

Summary