Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Convert degrees to radians.
- Use arc length with radians.
- Identify coterminal directions.
Before you start
Fractions and the circumference formula.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A marked radius on a radius-12 cm wheel turns counterclockwise through 150°. Express the turn in radians and find the distance travelled by its rim marker.
Why this math matters
Degrees count parts of a full turn. Radians compare curved distance with radius, making them especially useful when rotation must be connected to length.
Set up the model
A useful answer starts with clear assumptions:
- The wheel rotates about a fixed centre without changing radius.
- The rim marker follows the circular arc, not a straight chord.
- Counterclockwise is the positive direction.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Describe a wheel turn in degrees, radians, and arc length
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A marked radius on a radius-12 cm wheel turns counterclockwise through 150°. Express the turn in radians and find the distance travelled by its rim marker.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Compare with a complete turn
150/360 = 5/12 of a turn
This fraction is less than one half, consistent with the marked radius stopping before a half-turn.
Convert the angle
θ = (5/12)(2π) = 5π/6 radians
A complete turn measures 2π radians. Multiplying by the same turn fraction changes units without changing the rotation.
Measure the curved distance
s = rθ = 12(5π/6) = 10π cm ≈ 31.42 cm
The radius multiplies a radian measure. As a check, 5/12 of the 24π cm circumference is also 10π cm.
The result
The rotation is 5π/6 radians and the marker travels 10π cm.
A 510° rotation ends in the same direction but includes an extra revolution. Coterminal directions do not imply equal travelled distances.
Common mistakes to catch
- Using 150 directly in rθ treats degrees as radians.
- The straight chord between endpoints is shorter than this curved path.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Convert 225° to radians.
Show a hint
Multiply by π/180.
Reveal answer and explanation
5π/4
225π/180 simplifies by dividing numerator and denominator by 45.
Practice 2
A radius-4 m marker moves through π/3 radians. Find its arc length.
Show a hint
Multiply radius and angle.
Reveal answer and explanation
4π/3 m
s = 4(π/3), approximately 4.19 m.
Take the idea with you
Keep rotation amount, final direction, and travelled arc distance as separate quantities.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Choose the right ratio before touching a calculator
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