Radians and degree conversion | Year 11

01 / LEARN

The idea, step by step

A ratio of lengths

A radian angle is the arc length divided by the radius. One full turn therefore contains 2π radians. A radian is dimensionless; it is not a smaller version of one degree.

180^{\circ} = π rad

Convert before using a formula

Multiply degrees by $π / 180$ to obtain radians, or radians by $180 / π$ to obtain degrees. Cancel exact factors before approximating. Arc formulas require radians, and unmarked trigonometric function arguments are normally radians. Record the chosen mode when calculating numerical values.

A common trap

Treating one radian as one degree.

Instead: One radian is about 57.3°. Use the conversion factor rather than relabelling the number.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Convert 225° to radians.

$225 π / 180$ .
Cancel the common factor 45.
The result is in quadrant III, consistent with 225°.

5 π / 4

Example 2

Worked example 2

Convert $7 π / 6$ radians to degrees.

$(7 π / 6) (180 / π)$ .
Cancel pi;180/6=30.
Multiply 7 by 30.

210°

Check your understanding

Convert 90° to radians.

Show answer

$π / 2$

A right angle is a quarter turn.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Convert 150° to radians.
3 marks
Convert−60° to radians.
3 marks
Convert $11 π / 12$ 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 $π / 3$ .
3 marks
A calculator gives $sin 30 \approx - 0.988$ . Explain if the intended angle was 30°.
3 marks

Read the worked solutions

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

Convert 150° to radians.
3 marks
$150 π / 180 = 5 π / 6$ .
$5 π / 6$
Convert−60° to radians.
3 marks
$- 60 π / 180 = - π / 3$ .
$- π / 3$
Convert $11 π / 12$ radians to degrees.
3 marks
$(11 π / 12) (180 / π) = 165$ .
165°.
Convert 2.4 radians to degrees to one decimal place.
3 marks
$2.4 (180 / π) = 137.5098 \dots$ .
137.5°.
Give one positive and one negative angle coterminal with $π / 3$ .
3 marks
Add or subtract $2 π$ .
$7 π / 3$ and $- 5 π / 3$
A calculator gives $sin 30 \approx - 0.988$ . Explain if the intended angle was 30°.
3 marks
sin 30 radians≈−0.988, but sin 30°=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 ↗

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