Math lesson plan · 11th grade (Year 12 in England) · 45 minutes
Arc length and sector area in radians
Trigonometry
CCSS HSG.C.B.5England: A-level Pure Mathematics
Objective and success criteria
Students will: Arc length and sector area in radians.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Convert between degrees and radians; Arc length and sector area
Leads to: the next skill in this area.
Materials: Scientific calculators, Protractors and rulers, Right-triangle diagrams, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Convert between degrees and radians; Arc length and sector area). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Write 360° in radians, as a multiple of π. (3π/4 / 2π / 5π/4 / 5π/6) → 2πPrompt: 180° = π radians.
- A sector has radius 12 cm and angle 45°. Find the arc length in cm, rounded to 1 decimal place. → 9.4Prompt: Arc length = (θ/360) × 2πr.
- Convert π/6 radians to degrees. → 30Prompt: Replace π with 180°.
Hook · 5 min
- Show a ramp or a ladder against a wall and ask how steep it is.
- Share the objective: Arc length and sector area in radians. Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: A sector has radius 10 cm and angle 1.2 radians. Find its area.
- Step 1: ½ × 10² × 1.2 = 60 cm²
- Answer: 60
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- A sector has radius 9 cm and angle 0.4 radians. Find its arc length. → 3.6Prompt: In radians: arc length s = rθ and area A = ½r²θ.
- An arc of length 3 cm lies on a circle of radius 5 cm. Find the angle it makes at the centre, in radians. → 0.6Prompt: s = rθ, so θ = s ÷ r.
You do: independent practice · 10 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- A sector has radius 9 cm and angle 2 radians. Find its area. → 81Prompt: In radians: arc length s = rθ and area A = ½r²θ.
- An arc of length 10 cm lies on a circle of radius 4 cm. Find the angle it makes at the centre, in radians. → 2.5Prompt: s = rθ, so θ = s ÷ r.
Watch out for
- Mixing up opposite and adjacent sides.
- Calculator in radians instead of degrees.
Support and stretch
Support
- Review first: Convert between degrees and radians.
- Use concrete materials: scientific calculators.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-11.
Free lesson plan from eurekami.com/lesson-plan-builder