Until now you have measured angles in degrees, an arbitrary unit inherited from the Babylonians: $360$ equal parts for one turn, for no reason other than tradition. In Fifth Year Higher Level Maths a second unit appears, the radian, which is not arbitrary: it comes directly from comparing an arc of a circle with its radius. It is the unit used in physics and engineering because it simplifies the formulae for rotation, from the Ferris wheel at the Galway Christmas market to the disc of an angle grinder. By the end of this page you will be able to say how far a Luas wheel travels in one turn and how fast a hurling ball spins.
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 of the circle itself, as if it were a piece of string. The angle subtended by that piece of arc, exactly as long as the radius, is one radian. It does not matter whether the circle is small or enormous: the ratio between the arc and the radius is always the same for that angle.
Since the length of the whole circumference is $2\pi r$, and each radian corresponds to an arc of length $r$, exactly $2\pi$ radians fit into one full turn. That is why $2\pi$ rad is equivalent to $360^\circ$: it is not a coincidence, it is the very definition of the radian.
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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.
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 $\pi$).
One full turn is $360^\circ$ and also $2\pi$ rad. That equivalence is the rule of three that converts in either direction.
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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 $330^\circ$.
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 $\pi$.
Just as with degrees, an angle in radians can exceed $2\pi$ (more than one turn) or be negative (a clockwise rotation). To find where the angle "ends up", it is enough to subtract or add full turns of $2\pi$ rad until it lies between $0$ and $2\pi$.
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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 $-\frac{13\pi}{6}$.
This is where the radian proves its usefulness: since $1$ rad was defined as an arc equal to the radius, the length of any arc is simply the angle (in radians) multiplied by the radius, with no extra factor.
The same ideas work for arcs on the Earth: arc $=r\theta$ with $\theta$ in radians, and for clocks: the minute hand turns $6^\circ$ per minute and the hour hand $0.5^\circ$ per minute.
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Calculate the central angle subtended by an arc of length $12$ cm in a circle whose radius is $4$ cm. Express the result both in radians and in sexagesimal degrees.
Calculate the central angle subtended by an arc of length $12$ cm in a circle whose radius is $4$ cm. Express the result both in radians and in sexagesimal degrees.
On a circular running track, a runner travels along the inside lane, remaining $50$ metres from the centre. If they cover a distance of $120$ metres, calculate the angle…
On a circular running track, a runner travels along the inside lane, remaining $50$ metres from the centre. If they cover a distance of $120$ metres, calculate the angle described relative to the centre 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 $18$ metres 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 $18$ metres 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.
When something rotates (a disc, a wheel, the beam of a lighthouse), we want to know what angle it sweeps through per unit of time: that is the angular velocity, and since the natural angle for describing rotations is the radian, it is measured in radians per second.
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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.
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.
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.