Example 1
Circumference from radius
- Identify radius 6 cm.
- Find diameter 12 cm.
- Use C=πd.
- Calculate exact and approximate values.
Final answer: 12π cm, approximately 37.7 cm
Check: C=2πr also gives 12π cm.
What students learn in AC9M7M03
We are learning to describe relationships between features of circles, including radius, diameter and circumference, and use these relationships to solve problems.
Radius joins the centre to the circle, diameter spans the circle through its centre, and d=2r. Circumference measures the boundary length.
The constant ratio C÷d is π, giving C=πd=2πr. Exact answers retain π; decimal answers use ≈ and round only at the end.
Circumference models wheel turns and circular tracks, and the formulas can be rearranged to recover radius or diameter. Verify using the alternate equivalent formula.
Success criteria
- I can relate radius and diameter.
- I can calculate circumference exactly and approximately.
- I can rearrange a circumference formula and check the result.
Key vocabulary
- radius
- A line segment from the centre of a circle to its circumference.
- diameter
- A line segment across a circle through its centre, equal to twice the radius.
- circumference
- The distance around the boundary of a circle.
- pi
- The constant ratio of a circle’s circumference to its diameter, written π.
Visual models and representations
Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.
4 worked numerical & application examples
Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.
Example 2
Radius from circumference
- Use C=πd with C=31.4 m.
- Calculate d=31.4÷π≈10 m.
- Find r=d÷2.
- State the approximate radius.
Final answer: Approximately 5 m
Check: 2π(5)≈31.4 m.
Example 3
Application problem 1
Problem: A circular garden has circumference 31.4 m. Using π≈3.14, estimate its diameter and radius, then state the exact relationships used.
- Plan: Represent the information first, then calculate, interpret and independently check the result.
- Work: d = C/π = 10 m and r = 5 m. The relationships are C=πd and d=2r.
- Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Final answer: d = C/π = 10 m and r = 5 m. The relationships are C=πd and d=2r.
Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Example 4
Application problem 2
Problem: Compare circumferences of circles with radii 3 cm and 6 cm.
- Plan: Compare the radius scale factor.
- Work: The circumferences are 2π(3)=6π cm and 2π(6)=12π cm; because the radius doubles, the larger circumference is exactly twice the smaller.
- Interpret: The complete model for “Compare circumferences of circles with radii 3 cm and 6 cm.” shows The circumferences are 2π(3)=6π cm and 2π(6)=12π cm; because the radius doubles, the larger circumference is exactly twice the smaller.
Final answer: The circumferences are 2π(3)=6π cm and 2π(6)=12π cm; because the radius doubles, the larger circumference is exactly twice the smaller.
Check: The complete model for “Compare circumferences of circles with radii 3 cm and 6 cm.” shows The circumferences are 2π(3)=6π cm and 2π(6)=12π cm; because the radius doubles, the larger circumference is exactly twice the smaller.
Common misconceptions
Common mistake: Radius and diameter interchanged.
Correction: Label centre and endpoints.
Common mistake: π treated as an area unit.
Correction: It is a dimensionless ratio.
Common mistake: Circumference written in square units.
Correction: Use linear units.
10 important problems to solve
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. A circle has radius 4 cm. Find its diameter.
Check answer
Answer: 8 cm.
Hint: Double the radius.
Why: Using the method “Double the radius.” gives 8 cm.
2. A circle has diameter 9 m. Find its radius.
Check answer
Answer: 4.5 m.
Hint: Halve the diameter.
Why: Using the method “Halve the diameter.” gives 4.5 m.
3. Write circumference in terms of diameter.
Check answer
Answer: C=πd.
Hint: Multiply diameter by π.
Why: Using the method “Multiply diameter by π.” gives C=πd.
4. Find circumference for diameter 14 cm to one decimal place.
Check answer
Answer: 44.0 cm.
Hint: Keep π until the final rounding.
Why: Using the method “Keep π until the final rounding.” gives 44.0 cm.
5. Find circumference for radius 3.5 m to one decimal place.
Check answer
Answer: 22.0 m.
Hint: Use C=2πr.
Why: Using the method “Use C=2πr.” gives 22.0 m.
6. A wheel circumference is 2.2 m. How many turns cover 110 m?
Check answer
Answer: 50 turns.
Hint: Divide total distance by distance per turn.
Why: Using the method “Divide total distance by distance per turn.” gives 50 turns.
7. Is π exactly 3.14?
Check answer
Answer: No. The decimal 3.14 approximates the non-terminating irrational number π; retain π for an exact value and use ≈ for a rounded decimal.
Hint: Distinguish exact and approximate notation.
Why: The reasoning for “Is π exactly 3.14?” is complete because No. The decimal 3.14 approximates the non-terminating irrational number π; retain π for an exact value and use ≈ for a rounded decimal.
8. A circular track has circumference 400 m. Estimate diameter to one decimal place.
Check answer
Answer: Using C=πd gives d=400÷π≈127.3 m to one decimal place; checking π×127.3≈399.9 m, consistent after rounding.
Hint: Use d=C÷π.
Why: The complete model for “A circular track has circumference 400 m. Estimate diameter to one decimal place.” shows Using C=πd gives d=400÷π≈127.3 m to one decimal place; checking π×127.3≈399.9 m, consistent after rounding.
9. Compare circumferences of circles with radii 3 cm and 6 cm.
Check answer
Answer: The circumferences are 2π(3)=6π cm and 2π(6)=12π cm; because the radius doubles, the larger circumference is exactly twice the smaller.
Hint: Compare the radius scale factor.
Why: The complete model for “Compare circumferences of circles with radii 3 cm and 6 cm.” shows The circumferences are 2π(3)=6π cm and 2π(6)=12π cm; because the radius doubles, the larger circumference is exactly twice the smaller.
10. A circular garden has circumference 31.4 m. Using π≈3.14, estimate its diameter and radius, then state the exact relationships used.
Check answer
Answer: d = C/π = 10 m and r = 5 m. The relationships are C=πd and d=2r.
Hint: Represent the information first, then calculate, interpret and independently check the result.
Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Curriculum coverage and elaborations
Content description: describe the relationship between π and the features of circles including the circumference, radius and diameter.
International curriculum mapping
The Australian Curriculum code above is exact. Victorian Year 7, NSW Stage 4, US Grade 7, England Key Stage 3, New Zealand Level 4 and comparable international curricula contain broadly related learning, but code-to-code equivalence varies by jurisdiction.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Circles: What Is Pi?
Math Antics — Relate the diameter of a circle to its circumference using pi.
As you watch: What stays approximately constant when you divide circumference by diameter?
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Try it: Measure the circumference and diameter of two circular objects, calculate both ratios and discuss measurement error.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for The relationship between π and the features of circles including...
Mapped skill: describe the relationship between π and the features of circles including the circumference, radius and diameter
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M7M03 · Year 7 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M7M03 · Level 7 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA4-LEN-C-01 · Stage 4 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 7 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 7 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 8, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 7 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M7M03 — Circle Features and Circumference
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