Arc length and sector area | Year 11

01 / LEARN

The idea, step by step

Use radians

From the radian definition, $s = r θ$ . A sector is the fraction $θ / (2 π)$ of its circle, giving $K = r^{2} θ / 2$ . Convert degrees first and select the major or minor sector requested by the wording.

s = r θ, K = \frac{1}{2} r^{2} θ

Reverse problems and boundaries

Divide arc by radius to recover the angle. To recover radius from area, solve the equation and choose the positive root. A sector perimeter contains two radii as well as the curved arc. Keep units consistent and distinguish linear from square units.

A common trap

Substituting degrees into s=rθ.

Instead: Convert the angle to radians before applying this formula.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Find arc and area for radius 6 cm and angle $2 π / 3$ .

$s = 6 (2 π / 3) = 4 π$ .
$K = \frac{1}{2} (36) (2 π / 3) = 12 π$ .
Attach length and area units separately.

4 π cm

and

12 π {cm}^{2}

Example 2

Worked example 2

A sector has area 18 cm² and angle 1 radian. Find radius and perimeter.

$18 = r^{2} / 2$ , so $r = 6$ .
Arc length=6×1=6 cm.
Perimeter=6+2×6=18 cm.

Radius 6 cm; perimeter 18 cm.

Check your understanding

Find the arc for r=4 cm and θ=0.5 rad.

Show answer

2 cm.

s=4×0.5=2.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find arc length for r=10 cm, θ= $π / 5$ .
3 marks
Find sector area for r=5 m, θ=1.2 rad.
3 marks
Find the exact arc for radius 8 cm and angle 135°.
3 marks
A sector has radius 7 cm and arc 14 cm. Find angle and area.
3 marks
Find the full perimeter of a sector r=4 cm, θ= $π / 2$ .
3 marks
A 12 m² sector has radius 4 m. Find its angle in radians and degrees to one decimal place.
3 marks

Read the worked solutions

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

Find arc length for r=10 cm, θ= $π / 5$ .
3 marks
$s = 10 π / 5$ .
$2 π cm$
Find sector area for r=5 m, θ=1.2 rad.
3 marks
$K = 25 (1.2) / 2 = 15$ .
15 m².
Find the exact arc for radius 8 cm and angle 135°.
3 marks
$135^{\circ} = 3 π / 4$ ; multiply by 8.
$6 π cm$
A sector has radius 7 cm and arc 14 cm. Find angle and area.
3 marks
$θ = 14 / 7 = 2$ ; $K = 49 (2) / 2 = 49$ .
2 rad;49 cm².
Find the full perimeter of a sector r=4 cm, θ= $π / 2$ .
3 marks
Arc is $4 π / 2 = 2 π$ ; add both radii.
$8 + 2 π cm$
A 12 m² sector has radius 4 m. Find its angle in radians and degrees to one decimal place.
3 marks
$12 = 8 θ$ , so θ=1.5.
$1.5 (180 / π) = 85.9437 \dots$ .
1.5 rad≈85.9°.

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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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