MHF4U Radian Measure: Notes and Practice

Nothing in mathematics forces a circle to have $360$ parts. That number is a leftover from Babylonian astronomy — roughly one degree per day of the year, and $360$ happens to divide nicely. It works, but it is arbitrary, and every formula built on it drags an ugly conversion factor around. Radian measure throws the arbitrary number away and measures an angle using the circle itself.

What a radian actually is

Draw a circle of radius $r$ . Take a piece of string exactly $r$ units long and lay it along the circle's edge. The angle at the centre that this one-radius arc subtends is defined to be ONE RADIAN. Stretch the arc to a length $a$ and the angle grows in proportion, so in general $θ = \frac{a}{r}$ — an angle in radians is just the arc length measured in radii.

Fig 1One radian is the central angle whose arc is exactly as long as the radius.

How many radii fit around a whole circle? The circumference is $C = 2 π r$ , so exactly $2 π$ of them do. That single sentence is the entire reason one full revolution is $2 π$ radians, a half revolution is $π$ radians, and a quarter revolution is $\frac{π}{2}$ radians. Notice that $2 π \approx 6.283$ , so a radian is a fairly big angle — about $57.3°$ .

Radians have no units

A radian is a length divided by a length (arc over radius), so the units cancel and a radian is a pure number. That is why you may see

sin 2

with no symbol after the

2

— the bare number IS a radian measure. From this course onward, an angle written without a degree symbol is in radians by default.

Converting between degrees and radians

π radians = 180° ⟹ degrees \times \frac{π}{180} = radians, radians \times \frac{180}{π} = degrees

Both conversion factors are just the single fact on the left, rearranged.

Fig 2The same wheel of angles labelled twice: degrees outside, radians inside.

Going from degrees to radians, multiply by $\frac{π}{180}$ and then reduce the fraction — do NOT reach for a decimal. The expected form of an answer is exact, like $\frac{5 π}{6}$ .
Going from radians to degrees, multiply by $\frac{180}{π}$ ; the $π$ symbols cancel and you are left with a plain number of degrees.
A quick sanity check: $π \approx 3.14$ , so a radian answer bigger than about $6.28$ is more than one full turn, and an answer near $3.14$ is close to a half turn.

Degrees	Radians	Degrees	Radians
$0°$	$0$	$180°$	$π$
$30°$	$\frac{π}{6}$	$210°$	$\frac{7 π}{6}$
$45°$	$\frac{π}{4}$	$225°$	$\frac{5 π}{4}$
$60°$	$\frac{π}{3}$	$240°$	$\frac{4 π}{3}$
$90°$	$\frac{π}{2}$	$270°$	$\frac{3 π}{2}$
$120°$	$\frac{2 π}{3}$	$300°$	$\frac{5 π}{3}$
$135°$	$\frac{3 π}{4}$	$315°$	$\frac{7 π}{4}$
$150°$	$\frac{5 π}{6}$	$330°$	$\frac{11 π}{6}$

Worth knowing on sight. Every entry is a multiple of 30° or 45°, so apart from 0 and π every radian entry is a fraction of π with denominator 6, 4, 3, or 2.

VideoAn Ontario teacher's Grade 12 lesson on radian measure. Watch it once you have the definition and the conversion factor, to see both used on a run of examples.Watch on YouTube

Arc length and sector area

Fig 3The arc cut by a central angle, and the pizza-slice sector underneath it.

a = r θ A = \frac{1}{2} r^{2} θ where \frac{a}{2 π r} = \frac{A}{π r ^{2}} = \frac{θ}{2 π}

Each formula is just the angle's share of the whole circumference or the whole area. The share cancels the 2 pi cleanly only in radians, and both formulas require the angle in radians.

Radians are mandatory in these two formulas

If the angle arrives in degrees, convert it FIRST. Using

a = r θ

with

θ = 80

instead of

θ = 4 π /9

gives an answer roughly 57 times too big. The same trap lives in your calculator: set it to RAD mode for this entire unit, and check the mode indicator before every trig evaluation.

Finally, rotation over time. ANGULAR VELOCITY, usually written $ω$ (omega), is the angle swept per unit time: $ω = \frac{θ}{t}$ , measured in radians per second (or per minute). A rate given in revolutions per minute converts by multiplying by $2 π$ , since one revolution is $2 π$ radians. And because $a = r θ$ , dividing both sides by $t$ gives the linear speed of a point at radius $r$ : $v = r ω$ . A point on the rim of a spinning wheel and a point halfway in share the same $ω$ but not the same $v$ .

Radian Measure

Key things to remember

What a radian actually is

Converting between degrees and radians

Arc length and sector area

Worked examples

Where marks get lost

Flashcards 8

Practice