Radians and degree conversion | Year 11

01 / LEARN

Understand the idea.

A ratio of lengths

Radian measure compares the length of a circular arc with its radius. An arc as long as the radius subtends one radian. Because this is a ratio of two lengths, radian measure is dimensionless.

A full circumference is $2 π r$ , so dividing it by the radius gives $2 π$ radians for one complete turn. A half-turn therefore measures $π$ radians or $180^{\circ}$ , which supplies the conversion between the two scales.

180^{\circ} = π rad

Convert before using a formula

Convert by multiplying by a factor equal to one turn expressed in the two units. Degrees become radians using $\frac{π}{180}$ ; radians become degrees using $\frac{180}{π}$ . Cancel common factors to retain an exact multiple of $π$ .

Before using arc or sector formulas, convert to radians. Unmarked trigonometric arguments normally use radians too. Before calculating, match the calculator angle mode to the input: a correct number in the wrong mode represents a different angle.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Convert 225° to radians.

Worked solution

1
Multiply by the degree-to-radian factor.
$225^{\circ} = \frac{225 π}{180} rad$
A half-turn is both $180^{\circ}$ and $π$ radians.
2
Reduce the fraction.
$\frac{225 π}{180} = \frac{5 π}{4}$
Dividing numerator and denominator by 45 preserves the angle.
3
Check the turn size.
$π < \frac{5 π}{4} < \frac{3 π}{2}$
The angle is in quadrant III, consistent with $180^{\circ} < 225^{\circ} < 270^{\circ}$ .

Answer

$\frac{5 π}{4}$

EXAMPLE 02

Worked example 2

Convert $\frac{7 π}{6}$ radians to degrees.

Worked solution

1
Multiply by the radian-to-degree factor.
$\frac{7 π}{6} \times \frac{180^{\circ}}{π}$
The conversion uses $π rad = 180^{\circ}$ .
2
Cancel the common factors.
$\frac{7 π}{6} \times \frac{180^{\circ}}{π} = 7 (30^{\circ})$
The factor $π$ cancels and $180 \div 6 = 30$ .
3
State the angle in degrees.
$7 (30^{\circ}) = 210^{\circ}$
This is $30^{\circ}$ beyond a half-turn, matching the original radian angle.

Answer

210°

03 / YOUR TURN

Try it yourself.

Check your understanding

Convert 90° to radians.

Show answer & reasoning

$\frac{π}{2}$

$90^{\circ} \times \frac{π}{180^{\circ}} = \frac{π}{2}$ . This is one quarter of the full-turn measure $2 π$ .

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

Convert $150^{\circ}$ to radians.
3 marks
Convert $- 60^{\circ}$ to radians.
3 marks
Convert $\frac{11 π}{12}$ radians to degrees.
3 marks
Convert 2.4 radians to degrees to one decimal place.
3 marks
Give one positive and one negative angle coterminal with $\frac{π}{3}$ .
3 marks
A calculator gives $sin 30 \approx - 0.988$ . Explain if the intended angle was $30^{\circ}$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Convert $150^{\circ}$ to radians.
3 marks
$\frac{150 π}{180} = \frac{5 π}{6}$ .
$\frac{5 π}{6}$
Convert $- 60^{\circ}$ to radians.
3 marks
$- \frac{60 π}{180} = - \frac{π}{3}$ .
$- \frac{π}{3}$
Convert $\frac{11 π}{12}$ radians to degrees.
3 marks
$(\frac{11 π}{12}) (\frac{180}{π}) = 165$ .
$165^{\circ}$ .
Convert 2.4 radians to degrees to one decimal place.
3 marks
$2.4 (\frac{180}{π}) = 137.5098 \dots$ .
${137.5}^{\circ}$ .
Give one positive and one negative angle coterminal with $\frac{π}{3}$ .
3 marks
Add or subtract $2 π$ .
$\frac{7 π}{3}$ and $- \frac{5 π}{3}$
A calculator gives $sin 30 \approx - 0.988$ . Explain if the intended angle was $30^{\circ}$ .
3 marks
$sin (30 radians) \approx - 0.988$ , but $sin 30^{\circ} = 0.5$ .
The calculator was in radian mode.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

In this lesson

Understand the idea.

A ratio of lengths

Convert before using a formula

See the method in action.

Worked example 1

Multiply by the degree-to-radian factor.

Reduce the fraction.

Check the turn size.

Worked example 2

Multiply by the radian-to-degree factor.

Cancel the common factors.

State the angle in degrees.

Try it yourself.

Check your understanding

Keep practising