Lesson 11: Circumference of a Circle

§Text Passage

1Summary 重點摘要

Circumference formula: C = π × diameter = 2 × π × radius. π ≈ 3.14. Learn to calculate circumference and solve word problems.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: What is π?π (read as pi) is called the circular constant: for every circle, the circumference divided by the diameter gives the same number, about 3.14. π is actually an irrational number with endless, non-repeating decimals, but at primary level using 3.14 is accurate enough. Wrap a string round a round object, straighten it and measure: the length always comes to a little over 3 times the diameter; mathematicians in history used similar methods to obtain ever more precise values of π.
2. Block 2: Known DiameterWhen the diameter is known, use the formula C = πd. Example: a circle of diameter 10 cm has circumference 3.14 × 10 = 31.4 cm. Just multiply 3.14 by the diameter; remember to write the unit cm, because circumference is a length. Another example: diameter 4 cm gives circumference 3.14 × 4 = 12.56 cm; mentally, work out 314 × 4 = 1256 first, then move the decimal point two places left — quick and safe.
3. Block 3: Known RadiusWhen the radius is known, use C = 2πr. Example: a circle of radius 5 cm has circumference 2 × 3.14 × 5 = 31.4 cm. Its diameter is 10 cm, so the answer matches the previous one. The two formulas are really the same, because 2r is d. Another example: a circle of radius 3 cm has circumference 2 × 3.14 × 3 = 18.84 cm; work out 2 × 3 = 6 first, then multiply by 3.14 to get 18.84 with less effort.
4. Block 4: Known CircumferenceWhen the circumference is known, use the formula in reverse: d = C ÷ π. Example: if C = 31.4 cm, the diameter = 31.4 ÷ 3.14 = 10 cm, and dividing by 2 again gives radius 5 cm. If the division is not exact, keep the number of decimal places the question asks for. For example, if a wheel's circumference is 157 cm, the diameter = 157 ÷ 3.14 = 50 cm, and dividing by 2 again gives radius 25 cm; questions that give the circumference usually take these two steps.
5. Block 5: Semicircle PerimeterThe perimeter of a semicircle is not just the curved arc; the straight diameter must be added: semicircle perimeter = πr + 2r. Example: for radius 10 cm, perimeter = 3.14 × 10 + 2 × 10 = 31.4 + 20 = 51.4 cm. Forgetting the diameter is the most common error — a perimeter goes all the way round the boundary. Verify with radius 5 cm: semicircle perimeter = 3.14 × 5 + 2 × 5 = 25.7 cm, and the arc 15.7 is exactly half of the full circumference 31.4, so the figures echo one another.
6. Block 6: Word Problem: WheelThe distance a wheel covers in one turn equals its circumference. Example: a wheel of diameter 0.5 m moves 3.14 × 0.5 = 1.57 m in one turn, and 100 turns cover 1.57 × 100 = 157 m. This relation is often used to find rolling distances or the number of turns. Conversely, for a wheel of diameter 0.5 m (1.57 m per turn) that has rolled 314 m, the number of turns = 314 ÷ 1.57 = 200; distance and turns can be found from each other.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Circumference = πd = 2πr
• π ≈ 3.14
• Known diameter: C = πd
• Known radius: C = 2πr
• Semicircle: include the diameter