LEARNER JOURNEY Example 1: s=r
Set valuesLet s=5 cm and r=5 cm.
Form ratioθ=s/r.
Cancel unitsθ=(5 cm)/(5 cm)=1.
ConcludeThe central angle is 1 radian.
Start here
Build the concept from the beginning
Starting point
No prior trigonometry is assumed. Every required term and decision is taught on this page.
Learning target
Build the radian as a ratio of arc length to radius instead of treating π as a conversion trick.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- central angle
- An angle whose vertex is at the center of a circle.
- radian
- A unitless angle measure equal to arc length divided by radius.
- subtend
- To cut off or intercept an arc.
- ratio
- A quotient comparing two quantities; here θ=s/r.
Look first, then name the mathematics
Build the visual meaning
- radius r
- θ = 1 radian
- arc s = r
- s ÷ r = 1
Read the complete visual relationship as text
- The radius is r.
- The highlighted arc also has length r.
- Their ratio s/r is 1, so the angle is 1 radian.
Definition in plain language
For a central angle, θ = s/r when θ is measured in radians. If s = r, then θ = 1 radian. Because s and r use the same length unit, the units cancel.
Why this matters
Radians connect angles directly to circle geometry and make formulas such as s = rθ work without a conversion factor.
Reasoning path
See the method as a sequence
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Measure the intercepted arc length s.
Measure the radius r in the same units.
Compute θ = s/r.
Interpret the quotient as a radian measure and check whether the size is plausible.
The same method in words
- Measure the intercepted arc length s.
- Measure the radius r in the same units.
- Compute θ = s/r.
- Interpret the quotient as a radian measure and check whether the size is plausible.
Interactive decision model
Build the reasoning, one decision at a time
Construct a 2-radian situation.
Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.
The reasoning path changes as each decision is selected.
Choose every step, then check the construction.
Model before practice
Three fully explained examples
Each example names the decision, carries it out, and interprets the result.
Example 2: s=12 cm and r=4 cm
Identifys=12 and r=4.
Form ratioθ=12/4.
Calculateθ=3.
ConcludeThe angle is 3 radians, a little less than a half turn π.
Example 3: θ=0.5 rad and r=10 m
RearrangeFrom θ=s/r, use s=rθ.
Substitutes=10(0.5).
Calculates=5.
ConcludeThe intercepted arc is 5 m long, half the radius.
Practice with decreasing support
Guided practice
Practice immediately after the model
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. When is a central angle exactly 1 radian?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Which operation defines radian measure?
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Not completed yet.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Complete: if s=2r, then θ=____ radians.
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Independent practice
Six problems for repetition
Solve all six without opening the worked examples unless feedback shows what to revise.
1. Find θ when s=15 cm and r=5 cm.
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2. Find θ when s=4 m and r=8 m.
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3. Find s when r=6 and θ=2.
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4. Find r when s=18 and θ=3.
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5. Why do length units cancel in s/r?
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6. How should 3 radians be interpreted?
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Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. Which statement correctly compares 1 radian and 1 degree?
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2. If radius doubles while the same angle stays fixed, what happens to arc length?
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3. If s and r both double, what happens to θ?
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4. Find θ when s=7π and r=7.
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5. Why is θ=s·r incorrect?
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Unfamiliar transfer
Explain, compare, and revise
Explain why two circles of different sizes can show the same 1-radian angle, and give a numerical example.
Guided self-check
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Model reasoning
A radian depends on the ratio s/r, not on a particular length. On a circle with r=4 cm, an arc s=4 cm gives θ=4/4=1 radian. On a circle with r=10 cm, an arc s=10 cm gives θ=10/10=1 radian. The circles differ in size, but both ratios equal 1.
Write a first attempt before opening the self-check.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.