Angle Measure: Degrees and Radians
A snowboarder tells you she just landed a "1080." How many full spins is that? And how would a physicist describe the same rotation without using the number 360 at all? Degrees split a turn into $360$ pieces for historical reasons. In Integrated Math III you meet a second unit, the radian, which comes straight from the geometry of the circle. It is the unit that calculus, physics, and engineering use, because it makes the formulas for rotation much simpler.
The Radian: When the Arc Equals the Radius
Picture cutting a piece of string as long as the radius of a circle. Now lay that string along the edge of the circle. The angle at the center that "sees" that piece of arc is one radian. It doesn't matter whether the circle is a coin or a Ferris wheel: the angle is the same.
The whole circumference is $2\pi r$ long, and each radian takes up an arc of length $r$. So exactly $2\pi$ radians fit around a full circle. That is why $2\pi$ rad $= 360^\circ$. It is not a coincidence; it follows from the definition.
Since $\pi \approx 3.14$, a half turn is $\pi \approx 3.14$ rad, a quarter turn is about $1.57$ rad, and one radian is about $57.3^\circ$. These estimates let you place an angle without a calculator.
An angle measures exactly $3$ radians. Without using a calculator, reason whether this angle is greater than, less than or equal to a straight angle ($180^\circ$) and…
An angle measures exactly $3$ radians. Without using a calculator, reason whether this angle is greater than, less than or equal to a straight angle ($180^\circ$) and determine in which quadrant it is located.
Converting Between Degrees and Radians
A full turn is $360^\circ$ and also $2\pi$ rad. Dividing by $2$, a half turn is $180^\circ = \pi$ rad. That single equation lets you convert in either direction by setting up a proportion.
The same idea gives the interior angles of a polygon. The interior angles of an $n$-sided polygon add up to $(n-2)\cdot 180^\circ$. In a regular polygon all $n$ angles are equal, so each one measures $\dfrac{(n-2)\cdot 180^\circ}{n}$. For a regular hexagon the sum is $4\cdot180^\circ = 720^\circ$, and each angle is $120^\circ = \dfrac{2\pi}{3}$ rad.
Express in radians, using irreducible fractions of $\pi$, the following angles given in degrees: $15^\circ$, $135^\circ$, $210^\circ$ and $330^\circ$.
Express in radians, using irreducible fractions of $\pi$, the following angles given in degrees: $15^\circ$, $135^\circ$, $210^\circ$ and $330^\circ$.
Convert to degrees the following angles expressed in radians: $\frac{\pi}{5}$, $\frac{5\pi}{6}$, $\frac{7\pi}{4}$ and $\frac{11\pi}{3}$.
Convert to degrees the following angles expressed in radians: $\frac{\pi}{5}$, $\frac{5\pi}{6}$, $\frac{7\pi}{4}$ and $\frac{11\pi}{3}$.
Calculate the sum of the interior angles of a regular octagon and express the value of each of its interior angles in radians, using fractions of $\pi$.
Calculate the sum of the interior angles of a regular octagon and express the value of each of its interior angles in radians, using fractions of $\pi$.
Coterminal Angles and Quadrants
An angle in standard position has its vertex at the origin and its initial side along the positive $x$-axis. You rotate to the terminal side. Counterclockwise rotation is positive and clockwise rotation is negative. An angle can be larger than a full turn, or negative. Angles that end on the same terminal side are called coterminal.
Which quadrant is the terminal side in? Once the angle is between $0$ and a full turn, compare it to the quarter turns:
| Quadrant | Degrees | Radians |
|---|---|---|
| I | $0^\circ$ to $90^\circ$ | $0$ to $\frac{\pi}{2}$ |
| II | $90^\circ$ to $180^\circ$ | $\frac{\pi}{2}$ to $\pi$ |
| III | $180^\circ$ to $270^\circ$ | $\pi$ to $\frac{3\pi}{2}$ |
| IV | $270^\circ$ to $360^\circ$ | $\frac{3\pi}{2}$ to $2\pi$ |
Reduce the following angles to the first revolution (that is, find their equivalent angle between $0^\circ$ and $360^\circ$) and indicate in which quadrant the terminal…
Reduce the following angles to the first revolution (that is, find their equivalent angle between $0^\circ$ and $360^\circ$) and indicate in which quadrant the terminal side of each one is located: $1250^\circ$ and $2460^\circ$.
Find the equivalent positive angle in the first revolution (between $0$ and $2\pi$ rad) for the following negative angles: $-\frac{\pi}{4}$ and $-\frac{13\pi}{6}$.
Find the equivalent positive angle in the first revolution (between $0$ and $2\pi$ rad) for the following negative angles: $-\frac{\pi}{4}$ and $-\frac{13\pi}{6}$.
Arc Length and Area of a Sector
Here is where radians pay off. Since $1$ rad was defined as an arc equal to the radius, an arc that spans $\theta$ radians is $\theta$ radii long. No conversion factor is needed.
Calculate the central angle subtended by an arc of $12$ inches in length in a circle whose radius is $4$ inches. Express the result both in radians and in degrees.
Calculate the central angle subtended by an arc of $12$ inches in length in a circle whose radius is $4$ inches. Express the result both in radians and in degrees.
Determine the arc length described by the lower end of a pendulum of $60$ inches in length when it oscillates through an angle of $\frac{\pi}{8}$ radians.
Determine the arc length described by the lower end of a pendulum of $60$ inches in length when it oscillates through an angle of $\frac{\pi}{8}$ radians.
An agricultural irrigation sprinkler covers a circular sector. The water jet reaches a maximum distance of $18$ feet and the sprinkler head sweeps through an angle of…
An agricultural irrigation sprinkler covers a circular sector. The water jet reaches a maximum distance of $18$ feet and the sprinkler head sweeps through an angle of $140^\circ$. Calculate the total area of the watered land and the length of the outer arc described by the water.
Degrees–Minutes–Seconds and Distances on Earth
Maps and GPS use degrees split into smaller parts, just like hours are split into minutes and seconds: $1^\circ = 60'$ (arcminutes) and $1' = 60''$ (arcseconds). So $1^\circ = 3600''$.
The Earth is close to a sphere. Latitude measures an angle at the center of the Earth: $0^\circ$ at the equator, $90^\circ$ N at the North Pole. For two cities on the same meridian (a north–south line), the angle between them at the center is the difference of the latitudes if they are in the same hemisphere, and the sum if they are in opposite hemispheres. Then the distance along the surface is an arc: $s = r\theta$.
Eratosthenes' idea. When the sun's rays are parallel, the angle they make with a vertical stick in one city equals the central angle between the two cities (alternate interior angles). So if an arc of length $s$ on Earth matches angle $\theta$, then $r = \dfrac{s}{\theta}$, and the full circumference is $2\pi r$.
On a circular running track, a runner moves along the inner lane keeping $50$ feet from the center. If they cover a distance of $120$ feet, calculate the angle they have…
On a circular running track, a runner moves along the inner lane keeping $50$ feet from the center. If they cover a distance of $120$ feet, calculate the angle they have described with respect to the center of the track in radians. Also express this result in degrees, minutes and seconds.
Two cities, A and B, are located on the same terrestrial meridian. The latitude of city A is $41^\circ$ North and the latitude of city B is $28^\circ$ South. Assuming…
Two cities, A and B, are located on the same terrestrial meridian. The latitude of city A is $41^\circ$ North and the latitude of city B is $28^\circ$ South. Assuming that the Earth is a perfect sphere with a mean radius of $6371$ miles, calculate the straight-line distance on the Earth's surface that separates both cities.
In Eratosthenes' historic experiment, it was observed that in the city of Syene the sun's rays struck the ground perpendicularly (angle $0^\circ$) at noon on the summer…
In Eratosthenes' historic experiment, it was observed that in the city of Syene the sun's rays struck the ground perpendicularly (angle $0^\circ$) at noon on the summer solstice. At the same time, in Alexandria, located approximately $800$ miles north on the same meridian, the rays formed an angle of $\frac{\pi}{25}$ radians with the vertical. Exclusively from this data, deduce the radius of the Earth and the length of its meridian circumference.
Angular Velocity
When something spins, like a record, a tire, or a lighthouse beam, we care about how much angle it sweeps per unit of time. That is its angular velocity.
Often the speed is given in revolutions per minute (rpm). Convert in two separate steps: multiply by $2\pi$ (rev $\to$ rad), then divide by $60$ (min $\to$ s).
A vinyl record spins on a turntable at a speed of $33$ revolutions per minute (rpm). Calculate its angular velocity in radians per second and determine the angle, in…
A vinyl record spins on a turntable at a speed of $33$ revolutions per minute (rpm). Calculate its angular velocity in radians per second and determine the angle, in radians, described by a point on its edge in $2,5$ seconds.
The radius of the wheels of a car is $32$ inches. If the vehicle travels at a constant speed of $90$ mph, calculate the angular velocity of the wheels expressed in…
The radius of the wheels of a car is $32$ inches. If the vehicle travels at a constant speed of $90$ mph, calculate the angular velocity of the wheels expressed in radians per second. Subsequently, determine the total angle turned by a wheel in a journey of $15$ minutes and convert it to the number of complete revolutions.
A maritime lighthouse projects a beam of light that rotates at a constant angular velocity of $4$ revolutions per minute. The beam illuminates the surface of the sea up…
A maritime lighthouse projects a beam of light that rotates at a constant angular velocity of $4$ revolutions per minute. The beam illuminates the surface of the sea up to a maximum distance of $5$ miles. Due to the coastal topography, there is a blind sector of $\frac{2\pi}{5}$ radians inland where the light is not visible. Calculate the total area of the sea illuminated by the lighthouse in each complete revolution and the exact time in seconds that the light beam takes to cross the visible zone.
Rolling Wheels, Belts, and Gears
Many machines pass rotation from one part to another. The key idea is that the arc length is what gets shared, not the angle.
Rolling without slipping. If a wheel rolls on the ground, the distance its axle moves equals the arc length of wheel that has touched the ground: $d = r\theta$. One full turn moves it $2\pi r$.
Belts and meshing gears. In a belt that wraps two pulleys, or in two gears that mesh, the points on the rims move the same arc length. So $r_1\theta_1 = r_2\theta_2$, which means $\theta_2 = \theta_1\cdot\dfrac{r_1}{r_2}$. The smaller wheel turns through the larger angle. A belt keeps both pulleys turning the same direction. Meshing gears turn in opposite directions.
The transmission system of an industrial machine consists of two pulleys connected by a tight belt. The driving pulley has a radius of $15$ inches and the driven pulley…
The transmission system of an industrial machine consists of two pulleys connected by a tight belt. The driving pulley has a radius of $15$ inches and the driven pulley has a radius of $45$ inches. If the driving pulley rotates through an angle of $\frac{7\pi}{2}$ radians, calculate how many radians the driven pulley rotates and how many complete revolutions it makes.
A gear train is composed of three cogwheels A, B and C mounted in series, such that A meshes with B, and B meshes with C. The pitch radii of the wheels are $R_A = 12$…
A gear train is composed of three cogwheels A, B and C mounted in series, such that A meshes with B, and B meshes with C. The pitch radii of the wheels are $R_A = 12$ inches, $R_B = 8$ inches and $R_C = 20$ inches. If wheel A rotates through an angle of $150^\circ$ clockwise, determine the angle of rotation of wheel C in radians and specify its direction of rotation.
A cylindrical roller of radius $25$ inches is used to compact the asphalt of a street. When pushed, the roller advances in a straight line rolling without slipping and…
A cylindrical roller of radius $25$ inches is used to compact the asphalt of a street. When pushed, the roller advances in a straight line rolling without slipping and makes exactly $14,5$ complete revolutions. Calculate the horizontal distance traveled by the axis of the roller in feet and the total angle turned by any point on its surface, expressed in degrees.
Clocks, Polygons, and Sequences of Angles
An analog clock is a perfect model for angle problems. The minute hand turns $360^\circ$ in $60$ minutes, so it moves $6^\circ$ per minute. The hour hand turns $360^\circ$ in $12$ hours, which is $30^\circ$ per hour or $0.5^\circ$ per minute. Measure both hands clockwise from the $12$.
Two more angle ideas show up in this kind of problem. Interior angles of a polygon add to $(n-2)\cdot 180^\circ$. And when angles form an arithmetic sequence with common difference $d$, you can name the three angles of a triangle $a - d$, $a$, $a + d$. Their sum is $3a = 180^\circ$, so the middle angle is always $60^\circ$.
Calculate the smaller angle, expressed in degrees and in radians, formed by the hands of an analog clock (hour and minute hands) exactly at 4:20 p.m.
Calculate the smaller angle, expressed in degrees and in radians, formed by the hands of an analog clock (hour and minute hands) exactly at 4:20 p.m.
The three angles of a triangle are in arithmetic progression. If it is known that the largest angle measures $80^\circ$, determine the measure of the three angles…
The three angles of a triangle are in arithmetic progression. If it is known that the largest angle measures $80^\circ$, determine the measure of the three angles expressed in radians (using fractions of $\pi$).
Determine the exact time, between 7:00 and 8:00, when the hands of an analog clock (hour and minute hands) are perfectly superimposed. Also express the total angle swept…
Determine the exact time, between 7:00 and 8:00, when the hands of an analog clock (hour and minute hands) are perfectly superimposed. Also express the total angle swept by the hour hand from 7:00 to that precise moment, in radians.
Putting It Together: Equal Divisions, Equal Areas, and Infinite Series
The last few problems combine angle measure with algebra. Here are three patterns.
Equal spacing around a circle. If $n$ objects are spaced evenly around a circle, the angle between neighbors is $\dfrac{2\pi}{n}$, and the distance along the arc between them is $r\cdot\dfrac{2\pi}{n}$.
Matching areas. To compare sectors, set the area formulas equal and solve for the unknown angle. Start with a sector of radius $R$ and angle $\alpha$. Suppose a new sector has radius $2R$ and the same area. What angle $\beta$ does it need?
Infinite geometric series. A damped swing is a repeated process where each arc is a fixed fraction of the one before. If the first term is $a_1$ and the common ratio satisfies $|r| < 1$, the sum of all the terms is
$$S = \frac{a_1}{1 - r}.$$Back to the snowboarder. A "1080" is $\dfrac{1080}{360} = 3$ full spins, and in radians it is $1080\cdot\dfrac{\pi}{180} = 6\pi$ rad. It ends with the same terminal side as a $0^\circ$ angle, which is why she lands facing the same way she started.
A fairground Ferris wheel has a diameter of $40$ feet and has $24$ cabins spaced evenly along its outer edge. Calculate the distance along the arc of the Ferris wheel…
A fairground Ferris wheel has a diameter of $40$ feet and has $24$ cabins spaced evenly along its outer edge. Calculate the distance along the arc of the Ferris wheel that separates two consecutive cabins and the angle in radians they form with respect to the central axis.
There is an initial circular sector of radius $R$ and central angle $\alpha$ (in radians). A new circular sector is designed with the objective that its area is exactly…
There is an initial circular sector of radius $R$ and central angle $\alpha$ (in radians). A new circular sector is designed with the objective that its area is exactly equal to that of the original sector. If due to design restrictions the radius of the new sector must be three times the original ($3R$), determine algebraically what fraction of the angle $\alpha$ the new sector must have.
A pendulum of length $L = 1$ foot oscillates describing arcs of a circumference; each oscillation is the motion from one extreme position to the opposite one. In its…
A pendulum of length $L = 1$ foot oscillates describing arcs of a circumference; each oscillation is the motion from one extreme position to the opposite one. In its first oscillation, the maximum angle it forms with the vertical is $\frac{\pi}{6}$ radians. Due to friction, in each subsequent oscillation, the maximum angle reached is reduced by $10\%$ with respect to the immediately preceding oscillation. By setting up the sum of the corresponding infinite geometric series, calculate the total distance traveled by the mass of the pendulum until it stops completely.