Arc length and sector area

Visual model

A sector is a fraction of a full circle

\theta

arc length

r

\text{sector fraction}=\frac{\theta}{360}

A sector is a fraction of a full circle.

Use angle over 360.

Arc length uses circumference, sector area uses area.

Gold-standard guide

26 mins

What you will learn

Calculate parts of circles accurately.

Use a clear step-by-step method for arc length and sector area.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier geometry and measures skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Arc length = (θ/360) × 2* $\pi$ *r

Step 1

Calculate the arc length

Arc length = ( $15\frac{0}{3}60$ ) × 2 × $\pi$ × 12 = ( $\frac{5}{1}2$ ) × $24pi$ = $10pi c$ m

Step 2

Calculate the sector area

Sector area = ( $15\frac{0}{3}60$ ) × $\pi$ × 12² = ( $\frac{5}{1}2$ ) × $144pi$ = $60pi c$ m²

Watch out

Students use the diameter instead of the radius in the arc length formula, doubling their answer

Arc length

$arc = \thet\frac{a}{3}60 \times 2pi r.$

Sector area

$area = \thet\frac{a}{3}60 \times \pi r^{2}.$

Worked example

A sector has radius 12 cm and angle 150°. Find its arc length and area. Give answers in terms of $\pi$ .

Calculate the arc length: Arc length = ( $15\frac{0}{3}60$ ) × 2 × $\pi$ × 12 = ( $\frac{5}{1}2$ ) × $24pi$ = $10pi c$ m.

Calculate the sector area: Sector area = ( $15\frac{0}{3}60$ ) × $\pi$ × 12² = ( $\frac{5}{1}2$ ) × $144pi$ = $60pi c$ m².

Final answer

Arc length = $10pi c$ m; Sector area = $60pi c$ m²

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

A sector has radius 12 cm and angle 150°. Find its arc length and area. Give answers in terms of $\pi$ .

3 marks4 minsarc-length-and-sector-area-worked

Show solution

Worked solution

1.Calculate the arc length: Arc length = ( $15\frac{0}{3}60$ ) × 2 × $\pi$ × 12 = ( $\frac{5}{1}2$ ) × $24pi$ = $10pi c$ m.
2.Calculate the sector area: Sector area = ( $15\frac{0}{3}60$ ) × $\pi$ × 12² = ( $\frac{5}{1}2$ ) × $144pi$ = $60pi c$ m².

Final answer

Arc length = $10pi c$ m; Sector area = $60pi c$ m²

Mark points

M1: calculate the arc length
M1: calculate the sector area
A1: Arc length = $10pi c$ m; Sector area = $60pi c$ m²

Watch out

Students use the diameter instead of the radius in the arc length formula, doubling their answer.C = $2pi$ *r always — make sure r is correctly identified from the question before substituting.

Diagnosticrecall

Find the arc length of a sector with radius 8 cm and angle 45°. Give in terms of $\pi$ .

1 mark2 minsarc-length-and-sector-area-q1

Show solution

Worked solution

1.Spot the skill: Arc length = (θ/360) × 2* $\pi$ *r.
2.Use the calculate the arc length stage first, then calculate the sector area.
3.Keep the final answer visible: $2pi c$ m.

Final answer

$2pi c$ m

Mark points

M1: use the correct arc length = (θ/360) × 2* $\pi$ *r. sector area = (θ/360) × $\pi$ *r².the sector is a fraction of the full circle — multiply the full circumference or area by the fraction θ/360.
A1: $2pi c$ m

Watch out

Students use the diameter instead of the radius in the arc length formula, doubling their answer.C = $2pi$ *r always — make sure r is correctly identified from the question before substituting.

Easyprocedure

Find the area of a sector with radius 6 cm and angle 270°.

2 marks3 minsarc-length-and-sector-area-q2

Show solution

Worked solution

1.Spot the skill: Arc length = (θ/360) × 2* $\pi$ *r.
2.Use the calculate the sector area stage first, then calculate the arc length.
3.Keep the final answer visible: $27pi c$ m².

Final answer

$27pi c$ m²

Mark points

M1: use the correct arc length = (θ/360) × 2* $\pi$ *r. sector area = (θ/360) × $\pi$ *r².the sector is a fraction of the full circle — multiply the full circumference or area by the fraction θ/360.
A1: $27pi c$ m²

Watch out

Students use the diameter instead of the radius in the arc length formula, doubling their answer.C = $2pi$ *r always — make sure r is correctly identified from the question before substituting.

Mediumreasoning

A sector has arc length $5pi c$ m and radius 10 cm. Find the angle.

3 marks4 minsarc-length-and-sector-area-q3

Show solution

Worked solution

1.Spot the skill: Arc length = (θ/360) × 2* $\pi$ *r.
2.Use the calculate the arc length stage first, then calculate the sector area.
3.Keep the final answer visible: 90°.

Final answer

90°

Mark points

M1: use the correct arc length = (θ/360) × 2* $\pi$ *r. sector area = (θ/360) × $\pi$ *r².the sector is a fraction of the full circle — multiply the full circumference or area by the fraction θ/360.
A1: 90°

Watch out

Students use the diameter instead of the radius in the arc length formula, doubling their answer.C = $2pi$ *r always — make sure r is correctly identified from the question before substituting.

Hardproblem solving

Find the perimeter of a sector with radius 7 cm and angle 120°.

3 marks5 minsarc-length-and-sector-area-q4

Show solution

Worked solution

1.Spot the skill: Arc length = (θ/360) × 2* $\pi$ *r.
2.Use the calculate the sector area stage first, then calculate the arc length.
3.Keep the final answer visible: ( $14pi$ /3 + 14) cm.

Final answer

( $14pi$ /3 + 14) cm

Mark points

M1: use the correct arc length = (θ/360) × 2* $\pi$ *r. sector area = (θ/360) × $\pi$ *r².the sector is a fraction of the full circle — multiply the full circumference or area by the fraction θ/360.
A1: ( $14pi$ /3 + 14) cm

Watch out

Students use the diameter instead of the radius in the arc length formula, doubling their answer.C = $2pi$ *r always — make sure r is correctly identified from the question before substituting.

Exam-stylemulti-step

Find the area of a segment where r = 10 cm and angle = 60°.

4 marks6 minsarc-length-and-sector-area-q5

Show solution

Worked solution

1.Spot the skill: Arc length = (θ/360) × 2* $\pi$ *r.
2.Use the calculate the arc length stage first, then calculate the sector area.
3.Keep the final answer visible: area = ( $\pi$ × $10\frac{0}{6}$ ) − ( $\frac{1}{2}$ ×100×sin60°) = ( $50pi$ /3 − 25sqrt(3)) cm².

Final answer

area = ( $\pi$ × $10\frac{0}{6}$ ) − ( $\frac{1}{2}$ ×100×sin60°) = ( $50pi$ /3 − 25sqrt(3)) cm²

Mark points

M1: use the correct arc length = (θ/360) × 2* $\pi$ *r. sector area = (θ/360) × $\pi$ *r².the sector is a fraction of the full circle — multiply the full circumference or area by the fraction θ/360.
A1: area = ( $\pi$ × $10\frac{0}{6}$ ) − ( $\frac{1}{2}$ ×100×sin60°) = ( $50pi$ /3 − 25sqrt(3)) cm²

Watch out

Students use the diameter instead of the radius in the arc length formula, doubling their answer.C = $2pi$ *r always — make sure r is correctly identified from the question before substituting.

Grade 9 stretchproblem solving

A sector has radius 9 cm and angle 140 degrees. Find its arc length in terms of $\pi$ .

140^\circ

9\text{ cm}

arc

4 marks7 minssector-g9

Show solution

Worked solution

1.Use angle/360 of the full circumference.
2.Simplify the fraction.

Final answer

$7pi c$ m

Mark points

M1: $14\frac{0}{3}60$ × $18pi$
A1: $7pi c$ m

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Arc length and sector area - 2 marksFind the arc length of a sector with radius 8 cm and angle 45°. Give in terms of

\pi

.Mark answer

Answer

$2pi c$ m

2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer

Answer

3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer

Answer

Rectangle, square

4Perimeter, area and volume - 3 marksFind the perimeter of a rectangle with length 13 cm and width 8 cm.Mark answer

Answer

42 cm

Mastery check

I can explain the method for arc length and sector area.
I can show clear working without skipping key steps.
I can avoid this mistake: Students use the diameter instead of the radius in the arc length formula, doubling their answer.C = $2pi$ *r always — make sure r is correctly identified from the question before substituting.

Ask Nova Bot

Explain arc length and sector area from the beginning.Give me a hint for a arc length and sector area question.Quiz me on arc length and sector area.

Get help with anything that still feels tricky.

Ask Nova Bot

A sector is a fraction of a full circle

What you will learn

Know the rule, then use it

Method

Calculate the arc length

Calculate the sector area

Watch out

A sector has radius 12 cm and angle 150°. Find its arc length and area. Give answers in terms of π\pi π.

Build up to the hardest questions

Switch between skills

Get help with anything that still feels tricky.

A sector has radius 12 cm and angle 150°. Find its arc length and area. Give answers in terms of $\pi$ .