Year 7 Mathematics · AC9M7M03

Circle Features and Circumference

We are learning to describe relationships between features of circles, including radius, diameter and circumference, and use these relationships to solve problems

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Learning goalsSay it simply

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 conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Circumference from radius

  1. Identify radius 6 cm.
  2. Find diameter 12 cm.
  3. Use C=πd.
  4. Calculate exact and approximate values.

Final answer: 12π cm, approximately 37.7 cm

Check: C=2πr also gives 12π cm.

Example 2

Radius from circumference

  1. Use C=πd with C=31.4 m.
  2. Calculate d=31.4÷π≈10 m.
  3. Find r=d÷2.
  4. 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.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: d = C/π = 10 m and r = 5 m. The relationships are C=πd and d=2r.
  3. 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.

  1. Plan: Compare the radius scale factor.
  2. 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.
  3. 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.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7M03 - Circle Features and Circumference
Example 1

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.

Example 2

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

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

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.

Curriculum examplesCopied content

Content description: describe the relationship between π and the features of circles including the circumference, radius and diameter.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

Practice and reviewReady for practice

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.

Curriculum alignmentStart here

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.
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