Before you start, make sure you know: Angles, lengths, and areas (Secondary Year 4 (second cycle, academic track))
Wilkin looks at the Ferris wheel on the malecón and wonders: if a cabin moves along the rim, how can you tell how far it travels knowing only how much it turned? Until now you have measured angles in degrees, a unit that splits a full turn into $360$ parts purely by tradition. Now you will meet a second unit, the radian, which comes from the circle itself and makes the formulas for turns much simpler. At the end of the page we will come back to the wheel.
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 covers that piece of arc, exactly as long as the radius, is one radian. It doesn't matter whether the circle is tiny or huge: the arc/radius ratio for that angle is always the same.
The circumference measures $2\pi r$ and each radian covers an arc of length $r$, so exactly $2\pi$ radians fit in a full turn. That is why $2\pi$ rad $=360^\circ$.
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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 which quadrant it is in.
A full turn is $360^\circ$ and also $2\pi$ rad; a half turn is $180^\circ=\pi$ rad. With that pair you can set up a proportion.
Degrees, minutes, and seconds
When the result in degrees is not exact, it can be given in decimal degrees or in degrees, minutes, and seconds (sexagesimal notation). Each degree is divided into $60$ minutes and each minute into $60$ seconds: $1^\circ=60'$ and $1'=60''$.
To go back to decimal you do the opposite: $\text{degrees}+\dfrac{\text{minutes}}{60}+\dfrac{\text{seconds}}{3600}$.
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Express the following angles given in sexagesimal degrees in radians, using irreducible fractions of $\pi$: $15^\circ$, $135^\circ$, $210^\circ$ and $330^\circ$.
Express the following angles given in sexagesimal degrees in radians, using irreducible fractions of $\pi$: $15^\circ$, $135^\circ$, $210^\circ$ and $330^\circ$.
Many angle problems use two geometry facts you already know from Year 4, and you will need them in both degrees and radians.
The angle between the hands of a clock
The clock face has $12$ hours in $360^\circ$, so between two consecutive hours there are $30^\circ$. The minute hand makes one full turn in $60$ minutes, so it moves $6^\circ$ per minute. The hour hand moves $30^\circ$ in $60$ minutes, that is, $0.5^\circ$ per minute.
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Calculate the sum of the interior angles of a regular octagon and express the size 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 size of each of its interior angles in radians, using fractions of $\pi$.
The three angles of a triangle are in arithmetic progression. Knowing that the largest angle measures $80^\circ$, determine the size of the three angles expressed in…
The three angles of a triangle are in arithmetic progression. Knowing that the largest angle measures $80^\circ$, determine the size of the three angles expressed in radians (using fractions of $\pi$).
Calculate the smaller angle, expressed in sexagesimal degrees and in radians, formed by the hands of an analogue clock (hour hand and minute hand) at exactly 16:20.
Calculate the smaller angle, expressed in sexagesimal degrees and in radians, formed by the hands of an analogue clock (hour hand and minute hand) at exactly 16:20.
An angle can be more than one full turn or be negative (a clockwise turn). To find where it ends, you add or subtract full turns ($360^\circ$ or $2\pi$ rad) until it is between $0$ and $360^\circ$ (or between $0$ and $2\pi$).
The quadrants are numbered counterclockwise: I from $0^\circ$ to $90^\circ$, II from $90^\circ$ to $180^\circ$, III from $180^\circ$ to $270^\circ$, and IV from $270^\circ$ to $360^\circ$.
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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 lies: $1250^\circ$ and $2460^\circ$.
Find the equivalent positive angle in the first turn (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 turn (between $0$ and $2\pi$ rad) for the following negative angles: $-\frac{\pi}{4}$ and $-\frac{13\pi}{6}$.
On a circular running track, a runner travels along the inside lane, remaining $50$ meters from the center. If they cover a distance of $120$ meters, calculate the angle…
On a circular running track, a runner travels along the inside lane, remaining $50$ meters from the center. If they cover a distance of $120$ meters, calculate the angle described relative to the center of the track in radians. Also express this result in degrees, minutes, and seconds.
An agricultural irrigation sprinkler covers a circular sector. The water jet reaches a maximum distance of $18$ meters 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 $18$ meters and the sprinkler head sweeps through an angle of $140^\circ$. Calculate the total area of land irrigated and the length of the outer arc described by the water.
Go back to the beginning: if Wilkin's wheel has radius $r$ and the cabin turns $\theta$ radians, it travels an arc $s=r\theta$.
Extension: Angular Velocity
When something rotates (a disc, a wheel, the beam of a lighthouse), it is useful to know what angle it sweeps per unit of time: that is the angular velocity, measured in rad/s.
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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 drive system of an industrial machine consists of two pulleys connected by a taut belt. The driving pulley has a radius of $15$ cm and the driven pulley has a radius…
The drive system of an industrial machine consists of two pulleys connected by a taut belt. The driving pulley has a radius of $15$ cm and the driven pulley has a radius of $45$ cm. If the driving pulley turns through an angle of $\frac{7\pi}{2}$ radians, calculate how many radians the driven pulley turns and how many complete revolutions it makes.
The radius of the wheels of a car is $32$ cm. If the vehicle travels at a constant speed of $90$ km/h, calculate the angular speed of the wheels expressed in radians per…
The radius of the wheels of a car is $32$ cm. If the vehicle travels at a constant speed of $90$ km/h, calculate the angular speed of the wheels expressed in radians per second. Afterwards, determine the total angle turned by a wheel over a $15$-minute journey and convert it into the number of complete revolutions.