Angle Measure: Degrees and Radians

The Radian: When the Arc Equals the Radius

Degrees split a full turn into 360 equal pieces, which is a tradition and not a law of nature. The radian comes straight from the circle itself.

Imagine cutting a piece of string as long as the radius and laying it along the edge of the circle. The angle at the center that "sees" that piece of arc is one radian. Big circle or small circle, the angle is the same, because the arc and the radius grow together.

The whole circumference is $2\pi r$, so exactly $2\pi$ of those radius-long pieces fit around the circle. A full turn is $2\pi$ rad $= 360^\circ$, and a half turn is $\pi$ rad $= 180^\circ$. One radian is $\dfrac{180}{\pi}\approx 57.3^\circ$.

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive $x$-axis. Positive angles turn counterclockwise and negative angles turn clockwise. The side where the rotation stops is the terminal side. It's useful to know the benchmark values below, because they let you place an angle in radians without a calculator.

In decimals: $\dfrac{\pi}{2}\approx 1.57$, $\pi\approx 3.14$, $\dfrac{3\pi}{2}\approx 4.71$ and $2\pi\approx 6.28$.

Converting Between Degrees and Radians

A full turn is $360^\circ$ and also $2\pi$ rad. Cutting both in half gives the most convenient pair: $180^\circ = \pi$ rad. Set up a proportion with it.

Memorize these building blocks. Every other common angle is a multiple of one of them (for example, $120^\circ = 2\cdot 60^\circ = \tfrac{2\pi}{3}$, and $225^\circ = 5\cdot 45^\circ = \tfrac{5\pi}{4}$).

When an angle in radians is given as a plain number, there is no $\pi$ to cancel. For $2$ rad, $2\cdot\dfrac{180}{\pi}\approx 114.6^\circ$.

Degrees, minutes, seconds

Sometimes a fraction of a degree is written in minutes and seconds: $1^\circ = 60'$ and $1' = 60''$.

• Decimal to DMS: keep the whole degrees, multiply the decimal part by $60$ to get minutes, then multiply the new decimal part by $60$ to get seconds. For example, $28.35^\circ$: $0.35\cdot 60=21$, so $28^\circ 21'$.

• DMS to decimal: $40^\circ 15' = 40+\tfrac{15}{60}=40.25^\circ$.

For $2.4$ rad: $2.4\cdot\dfrac{180}{\pi}\approx 137.5099^\circ$. Then $0.5099\cdot 60\approx 30.59'$ and $0.59\cdot 60\approx 35.5''$. So $2.4\text{ rad}\approx 137^\circ 30' 36''$.

Angles in Polygons and Triangles

Angles in geometry still count, and in precalculus you'll often be asked for them in radians. The key fact: a polygon with $n$ sides can be split into $n-2$ triangles, drawing diagonals from one vertex. Each triangle contributes $180^\circ=\pi$ rad.

The angles of any triangle add to $180^\circ=\pi$ rad. If three angles form an arithmetic sequence, you can write them as $a,\ a+d,\ a+2d$, where $d$ is the common difference. Some problems call this an arithmetic progression; it's the same thing.

Clock Problems

A clock is a circle, so it's a good place to practice angles. Measure each hand's angle clockwise from the 12.

• The minute hand makes a full turn ($360^\circ$) in $60$ min, so it moves $6^\circ$ per minute.

• The hour hand makes a full turn in $12$ h, so it moves $30^\circ$ per hour, which is $0.5^\circ$ per minute.

At $h$ hours and $m$ minutes: minute hand $=6m$ degrees, hour hand $=30h+0.5m$ degrees. The angle between them is the difference of the two. If it's bigger than $180^\circ$, subtract it from $360^\circ$ to get the smaller angle.

When do the hands line up? Set the two angle formulas equal. Between 2:00 and 3:00, let $t$ be the minutes after 2:00. The minute hand is at $6t$ and the hour hand at $60+0.5t$:

The hands overlap at about $2{:}10{:}55$. The minute hand gains $5.5^\circ$ per minute on the hour hand, and that "relative speed" is the heart of every clock problem.

Coterminal Angles and Quadrants

Two angles in standard position are coterminal if they have the same terminal side. Adding or subtracting full turns doesn't change where the angle ends.

For big angles in radians, ask how many copies of $2\pi$ fit. For $\dfrac{25\pi}{4}$: $2\pi=\dfrac{8\pi}{4}$, and three copies make $\dfrac{24\pi}{4}$. So $\dfrac{25\pi}{4}-\dfrac{24\pi}{4}=\dfrac{\pi}{4}$.

Arc Length

Now the radian pays off. Since $\theta=\dfrac{s}{r}$ by definition, solving for $s$ gives the arc length with no extra factors.

You can also solve for the angle: $\theta=\dfrac{s}{r}$. For an arc of $15$ ft on a circle of radius $6$ ft, $\theta=\dfrac{15}{6}=2.5$ rad, which is $2.5\cdot\dfrac{180}{\pi}\approx143.2^\circ$.

Area of a Sector

A sector is the "pizza slice" of a circle: two radii and the arc between them. Its area is the same fraction of the whole circle as its angle is of a full turn.

Sometimes the area is given and you need the angle. A sector with area $30\text{ ft}^2$ and radius $5$ ft has $\theta=\dfrac{2A}{r^2}=\dfrac{60}{25}=2.4$ rad.

Arcs on Earth and on a Track

Any distance along a curve can be read as an arc: a runner on a circular track, a ship following a meridian, a lighthouse beam. When you know two of $s$, $r$, $\theta$, the third follows from $s=r\theta$.

Distances on Earth. Treat Earth as a sphere. Two cities on the same meridian (line of longitude) lie on the same great circle, so the arc between them has a central angle equal to the difference in their latitudes:

• opposite hemispheres (one North, one South): add the latitudes;

• same hemisphere: subtract them.

Finding the radius from an arc. You can run the formula backwards: $r=\dfrac{s}{\theta}$, and the circumference is $2\pi r$.

On a track. If a runner covers $s$ feet along a lane of radius $r$, the angle swept around the center is $\theta=\dfrac{s}{r}$ rad. To express an answer in degrees, minutes, and seconds, convert with $\dfrac{180}{\pi}$ and then split the decimals as in the DMS section.

Rolling Wheels, Pulleys, and Gears

A wheel rolling without slipping. The part of the rim that touches the ground lays itself down on the ground. So when the wheel turns through $\theta$ radians, its center moves forward a distance $s=r\theta$.

If the wheel makes $n$ full revolutions, $\theta=2\pi n$ rad, so $s=2\pi r n$: each revolution advances one circumference.

Belts and gears. When two wheels are linked by a belt or their teeth mesh, a point on one rim moves exactly as far as a point on the other. The arc lengths are equal:

The smaller wheel turns through the bigger angle. The direction depends on the link:

• belt (open belt): both pulleys turn the same way;

• meshing gears: neighbors turn in opposite directions.

In a chain $A\to B\to C$ the middle wheel $B$ flips the direction twice, so $C$ turns the same way as $A$. Because $r_A\theta_A=r_B\theta_B=r_C\theta_C$, the middle radius cancels out of the final answer.

Angular Speed

When something spins, we care about how much angle it sweeps per unit of time. That's angular speed, $\omega$ (omega), usually in radians per second.

Revolutions per minute (rpm). One revolution is $2\pi$ rad. Go in two steps: multiply by $2\pi$ to get rad/min, then divide by $60$ to get rad/s. To go back, $n$ revolutions correspond to $\theta=2\pi n$.

Linear speed units. To use $v=\omega r$ with a car or bike speed, make the units agree. Useful facts: $1\text{ mi}=5{,}280$ ft $=63{,}360$ in., and $60\text{ mph}=88$ ft/s.

Back to the start: a radian is the angle whose arc equals the radius, which makes formulas like $s=r\theta$ and $v=\omega r$ free of conversion factors, and that's why calculators and science use radians. And yes, the seat on the rim of a Ferris wheel moves faster than a seat near the hub, because every seat has the same $\omega$ but a different $r$.

Shrinking Arcs: A Geometric Series

A swinging pendulum loses energy, so each swing covers a smaller angle. If the angle shrinks by the same percent every time, the arc lengths form a geometric sequence, and the total distance is an infinite geometric series.

For a pendulum of length $L$, the arc of a swing with angle $\theta_k$ is $s_k=L\theta_k$. If each angle is $r$ times the one before ($r$ is the common ratio), then

A decrease of $p\%$ each time multiplies the angle by $r=1-\dfrac{p}{100}$.

Summary