LEARNER JOURNEY Example 1: circumference with r = 4 cm
Identifyr = 4 cm.
FormulaC = 2πr.
SubstituteC = 2π(4 cm).
ConcludeC = 8π cm.
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
Connect radius, diameter, circumference, and arc length before defining radians.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- center
- The point equally distant from every point on the circle.
- radius r
- A segment from the center to the circle.
- diameter
- A segment across the circle through the center; length 2r.
- circumference
- The distance around the entire circle; 2πr.
- arc length s
- The distance along a selected portion of the circumference.
Look first, then name the mathematics
Build the visual meaning
- Center
- Radius r
- Diameter 2r
- Arc
- Arc length s
- Circumference = 2πr
Read the complete visual relationship as text
- A radius runs from center to circle.
- A diameter crosses the center and equals two radii.
- An arc is part of the circumference.
- Arc length s measures that curved part.
Definition in plain language
For a central angle θ measured in radians, arc length is s = rθ. For a fraction f of a full turn, s = f(2πr).
Why this matters
Arc length gives radians their meaning: a radian compares the curved distance s with the radius r.
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.
Identify the radius r and the fraction of the full circle, or the radian angle θ.
Use C = 2πr for a full circumference.
Use s = fC for a fractional turn or s = rθ for radians.
Keep length units on s and no length unit on θ.
The same method in words
- Identify the radius r and the fraction of the full circle, or the radian angle θ.
- Use C = 2πr for a full circumference.
- Use s = fC for a fractional turn or s = rθ for radians.
- Keep length units on s and no length unit on θ.
Interactive decision model
Build the reasoning, one decision at a time
Construct the arc-length setup for a semicircle with diameter 14 cm.
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: quarter-circle arc with r = 6 m
FractionA quarter circle has f = 1/4.
CircumferenceC = 2π(6) = 12π m.
Take fractions = (1/4)(12π).
Concludes = 3π m.
Example 3: arc for θ = 2 radians and r = 3 ft
Identifyr = 3 ft and θ = 2.
Formulas = rθ.
Substitutes = 3(2).
Concludes = 6 ft.
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. What segment runs from the center to the circle?
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2. How is diameter related to radius?
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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 the circumference formula: C = ____.
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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 the circumference when r=5.
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2. Find a half-circle arc length when r=5.
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3. Find a quarter-circle arc length when r=8.
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4. If diameter is 18, what is the radius?
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5. Use s=rθ: r=7 and θ=3. Find s.
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6. Use s=rθ: s=20 and r=5. Find θ.
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Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. What type of unit belongs on an arc length?
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2. Find C if the diameter is 12.
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3. A 90° arc is what fraction of a circle?
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4. Find the 90° arc length when r=10.
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5. Why is s=2πr wrong for a 60° arc?
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Unfamiliar transfer
Explain, compare, and revise
A wheel has diameter 24 inches and turns through one third of a revolution. Find the distance traveled along its rim and explain every quantity used.
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
The diameter is 24 in, so the circumference is C = πd = 24π in. One third of a revolution traces one third of the circumference: s = (1/3)(24π) = 8π in. The answer is not 24π because the wheel traces only one third of a full turn.
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.