Measure the Boundary of a Circle
A circle’s circumference is π times its diameter, equivalently 2π times its radius.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: A model circle has radius 2 cm. Use π≈3.14 to estimate circumference. Candidate answers: (A) 12.56 cm (B) 14.56 cm (C) 4 cm (D) 6.28 cm.
- Independent check inputs: A model circle has radius 3 cm. Use π≈3.14 to estimate circumference. Candidate answers: (A) 18.84 cm (B) 28.26 cm (C) 6 cm (D) 9.42 cm.
Learn the idea
Geometric measures answer different questions: circumference is boundary length, area covers a region, and volume counts space. A formula works only when its input dimensions and output units match the question being asked. A circle’s circumference is π times its diameter, equivalently 2π times its radius. The factor two converts a center-to-boundary radius into a full across-circle diameter.
Here is the method on the supplied teaching case. Circumference is 2πr. The diameter is 4 cm, so the estimate is 3.14×4=12.56 cm. This is length around, not area inside.
Do not substitute the radius into πd without first doubling it. Here π≈3.14 is an instructed approximation; a circumference should carry length units rather than squared units.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: A model circle has radius 2 cm. Use π≈3.14 to estimate circumference. Candidate answers: (A) 12.56 cm (B) 14.56 cm (C) 4 cm (D) 6.28 cm. Solution: Circumference is 2πr. The diameter is 4 cm, so the estimate is 3.14×4=12.56 cm. This is length around, not area inside. The final answer is 12.56 cm. Do not substitute the radius into πd without first doubling it. Here π≈3.14 is an instructed approximation; a circumference should carry length units rather than squared units.
Check your reasoning
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Helpful tips
- A circle’s circumference is π times its diameter, equivalently 2π times its radius.
- Do not substitute the radius into πd without first doubling it.
See the answer and explanation
18.84 cm
Circumference is 2πr. The diameter is 6 cm, so the estimate is 3.14×6=18.84 cm. This is length around, not area inside.
Take it further
Changed-input task: A model circle has radius 9 cm. Use π≈3.14 to estimate circumference. Candidate answers: (A) 18 cm (B) 254.34 cm (C) 28.26 cm (D) 56.52 cm. Show the mathematical reason, not only the selected answer. Expected result: 56.52 cm. Circumference is 2πr. The diameter is 18 cm, so the estimate is 3.14×18=56.52 cm. This is length around, not area inside.
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