Define radian measure and convert degrees and radians

Ontario Grade 12 Mathematics

Trigonometric Functions

Ontario Grade 12 Mathematics — B1.1

Angles can be measured in degrees or radians. A degree divides a full turn into 360 equal parts. A radian measures an angle by comparing the length of its arc with the circle’s radius. This comparison gives a fixed relationship between the two units, which makes conversion straightforward. We begin with familiar circle ideas, define radian measure, and then practise converting in both directions.

What you will learn

Define radian measure using the radius and arc length of a circle.
Explain the relationship between degrees and radians.
Convert angles from degrees to radians and from radians to degrees.

1. Review: angles, radius, and arcs

An angle describes a turn between two rays that begin at the same point. One full turn is

360^\circ

. A half turn is

180^\circ

, and a quarter turn is

90^\circ

. The raised circle is the degree symbol.

A circle’s radius is the distance from its centre to its edge. An arc is a curved part of the circle’s edge. When an angle’s vertex is at the centre, its two rays meet the circle at two points and enclose an arc. On the same circle, a larger central angle cuts off a longer arc.

A quarter turn cuts off one quarter of the circle’s circumference. Circumference means the distance around a circle. These ideas let us describe an angle using the arc it makes, rather than counting degree-sized parts.

C=2\pi r

A central angle has its vertex at the circle’s centre.
The radius runs from the centre to the circle’s edge; an arc is part of that edge.
Degrees measure turns by dividing a full turn into equal parts.

2. What radian measure means

Radian measure describes an angle by comparing its arc length with the radius. Arc length is the distance along the curved arc, not the straight distance between its endpoints. Use a circle centred at the angle’s vertex. Let

s

be the arc length and

r

the radius. The angle’s radian measure is the ratio of these lengths.

The ratio stays the same if you use a larger or smaller circle for the same angle. When the circle grows, both its radius and the arc length grow by the same factor. Radian measure therefore depends on the angle, not on the circle chosen to illustrate it.

An angle measures one radian when its arc length equals the radius. If its arc length is twice the radius, its measure is two radians. Radians are often written without a unit symbol, though writing “rad” can clarify the unit.

For a full turn, the arc is the entire circumference. Dividing the circumference by the radius gives the full-turn measure in radians. The same full turn is

360^\circ

. This relationship gives the main fact used in conversions.

\theta=\frac{s}{r}

Radian measure is arc length divided by radius.
An angle of one radian cuts off an arc whose length equals the radius.
A full turn measures $2\pi$ radians, the same angle as $360^\circ$ .

3. Convert between degrees and radians

Half of a full turn is

180^\circ

and also

\pi

radians. So

180^\circ=\pi

radians. This is the central conversion fact.

To convert degrees to radians, multiply the degree measure by

\pi/180

. To convert radians to degrees, multiply by

180/\pi

. These factors reverse one another. Keeping the starting unit in view helps you choose the correct factor.

Leave answers involving

\pi

in exact form unless a decimal is requested. Familiar equivalents can help check your work:

90^\circ=\pi/2

,

180^\circ=\pi

, and

360^\circ=2\pi

. For example, an angle smaller than

180^\circ

should have a radian measure smaller than

\pi

.

\text{radians}=\text{degrees}\times\frac{\pi}{180}

Degrees to radians: multiply by $\pi/180$ .
Radians to degrees: multiply by $180/\pi$ .
The two conversion factors come from the equivalence $180^\circ=\pi$ radians.

Common degree–radian equivalents

Degrees	Radians	Turn description
$0^\circ$	$0$	No turn
$90^\circ$	$\pi/2$	Quarter turn
$180^\circ$	$\pi$	Half turn
$270^\circ$	$3\pi/2$	Three-quarter turn
$360^\circ$	$2\pi$	Full turn

Worked example

Convert in both directions

Convert

135^\circ

to radians. Then convert

\frac{7\pi}{6}

radians to degrees.

Convert degrees to radians
Multiply the degree measure by $\pi/180$ . The conversion factor uses the equivalence between $180^\circ$ and $\pi$ radians.
$135\times\frac{\pi}{180}$
Simplify the fraction
Reduce $135/180$ by dividing the numerator and denominator by $45$ . Keep $\pi$ to give an exact answer.
$\frac{135\pi}{180}=\frac{3\pi}{4}$
Convert radians to degrees
Multiply by $180/\pi$ . The factors of $\pi$ cancel, leaving a measure in degrees.
$\frac{7\pi}{6}\times\frac{180}{\pi}=210$

Answer:

135^\circ=\frac{3\pi}{4}

radians, and

\frac{7\pi}{6}

radians is

210^\circ

.

Check: The first angle is less than

180^\circ

, and

3\pi/4

is less than

\pi

. The second angle is greater than

\pi

radians, so its degree measure is greater than

180^\circ

. Both comparisons fit the results.

Common mistakes and how to avoid them

Multiplying by

180/\pi

when converting degrees to radians.

Correction: Use

\pi/180

for degrees to radians. Use the reverse factor,

180/\pi

, for radians to degrees.

Treating arc length as the straight distance between the arc’s endpoints.

Correction: Arc length follows the circle’s curved edge. Radian measure uses this curved length divided by the radius.

Assuming every radian measure must be written with a “rad” label or contain

\pi

.

Correction: Radians are often written without a unit label, and some radian measures are not multiples of

\pi

. Use the context to identify the unit.

Rounding a value involving

\pi

too early.

Correction: Keep the exact form with

\pi

during the conversion. Round only if the question asks for a decimal approximation.

Lesson summary

Radian measure is the ratio of arc length to radius.
One full turn is $360^\circ=2\pi$ radians, so a half turn is $180^\circ=\pi$ radians.
Multiply a degree measure by $\pi/180$ to convert to radians.
Multiply a radian measure by $180/\pi$ to convert to degrees.
Keep exact answers in terms of $\pi$ unless an approximation is requested.

Check your understanding

Question 1

What is the radian measure of

60^\circ

?

$\frac{\pi}{3}$
$\frac{\pi}{6}$
$\frac{3\pi}{2}$
$\frac{5\pi}{3}$

Show answer and explanation

\frac{\pi}{3}

Multiply by

\pi/180

:

60\times\pi/180=\pi/3

.

Question 2

Convert

\frac{5\pi}{4}

radians to degrees.

$135^\circ$
$225^\circ$
$250^\circ$
$315^\circ$

Show answer and explanation

225^\circ

Multiply by

180/\pi

:

(5\pi/4)(180/\pi)=225^\circ

.

Question 3

A central angle cuts off an arc of length

9

units in a circle with radius

3

units. What is its radian measure?

$3$ radians
$6$ radians
$\frac{1}{3}$ radian
$27$ radians

Show answer and explanation

3

radians

Radian measure is arc length divided by radius, so

9/3=3

radians.

Key terms

Arc: A curved part of a circle’s edge.
Arc length: The distance measured along an arc.
Central angle: An angle whose vertex is at the centre of a circle.
Radian measure: An angle measure equal to the arc length divided by the radius of the circle.
Radius: The distance from the centre of a circle to its edge.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B1.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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