Circumference (intro)
FoundationCircumference = diameter × π (~3.14).
After this lesson, you can:
- Circumference: C = πd or C = 2πr
- π ≈ 3.14, the ratio circumference ÷ diameter
- Diameter = 2 × radius
1Summary 重點摘要
Circumference = diameter × π (~3.14).
2Vocabulary cards Vocabulary
C
Circle圓
A shape whose points are all equidistant from the centre.
C
Centre圓心
The middle point of the circle, usually called O.
R
Radius半徑（r）
The distance from centre to edge, e.g. 5 cm.
D
Diameter直徑（d）
A chord through the centre touching both sides; d = 2r.
C
Circumference圓周
The distance all the way around the circle.
P
Pi (π)圓周率（π）
Circumference ÷ diameter for ANY circle, about 3.14.
P
Perimeter周界
Total boundary length; for a circle it is the circumference.
S
Semicircle半圓
Half a circle; perimeter = πr + d (arc plus diameter).
3Step by step Steps
- Step 1: Parts of a circleThe middle point is the centre O; the radius r runs from the centre to the edge; the diameter d passes through the centre with both ends on the circle. A circle of radius 5 has diameter 10, because d = 2r. Knowing these parts keeps the formulas straight.
- Step 2: The two formulasWith the diameter use C = πd; with the radius use C = 2πr. They are really one formula, because putting d = 2r into πd gives 2πr. Pick whichever matches what you know.
- Step 3: What π isMeasure the circumference and the diameter of any round object, such as a tin or a wheel, and divide: the answer is always about 3.14. This fixed ratio is π, the same for every circle.
- Step 4: Using the diameterFor d = 10 cm: C = 3.14 × 10 = 31.4 cm, shifting the decimal point one place right. The circumference should be a little more than three diameters, and 31.4 fits that expectation.
- Step 5: Using the radiusFor r = 3 cm: C = 2 × 3.14 × 3. First 2 × 3.14 = 6.28, then 6.28 × 3 = 18.84 cm. Working out 2π first can be reused whenever the radius changes.
- Step 6: Semicircle perimeterFor a semicircle of d = 10: the arc is half the circle, 3.14 × 10 ÷ 2 = 15.7, and the straight diameter must be added: 15.7 + 10 = 25.7 cm. Counting only the arc and forgetting the straight edge is the classic mistake.
4Diagrams Diagrams
5Worked example Example
Example: a circular garden has diameter 10 m. How long is one lap around it? (Use π = 3.14)
- The diameter is given, d = 10 m, so choose the formula C = πd.
- Substitute: C = 3.14 × 10.
- Calculate: 3.14 × 10 = 31.4, shifting the point one place right.
- Attach the unit: the circumference is 31.4 m.
- Sense check: one lap should be a bit over three diameters, and 31.4 ÷ 10 = 3.14 matches π.
- Answer: One lap around the garden is about 31.4 m. With diameter 10 m the formula C = πd gives 3.14 × 10 = 31.4. As a check, the ratio of circumference to diameter is 31.4 ÷ 10 = 3.14, which is exactly π, so the calculation agrees with the meaning of circumference.
6Common mistakes Mistakes
✗ Using π = 3: C = 3 × 10 = 30.
✓ At primary level π ≈ 3.14: 3.14 × 10 = 31.4.
✗ Mixing radius with diameter: taking r = 5 as d.
✓ Sort it out first: d = 2r, so r = 5 means d = 10.
✗ Swapping circumference with area: using πr² for the perimeter.
✓ Circumference is 2πr or πd; πr² is area (next lesson).
✗ Counting only the arc for a semicircle.
✓ Semicircle perimeter = arc + diameter: 15.7 + 10 = 25.7.
7Quick check Quick check
Q1Diameter 10 cm. Circumference? (π = 3.14)
Answer: 3.14 × 10 = 31.4 cm.
Did you get it right?
Q2Radius 4 cm. Circumference?
Answer: 2 × 3.14 × 4 = 25.12 cm.
Did you get it right?
Q3What is π approximately?
Answer: 3.14 (circumference ÷ diameter of any circle).
Did you get it right?
Q4What is the relation between radius and diameter?
Answer: Diameter = 2 × radius.
Did you get it right?
Q5Perimeter of a semicircle with diameter 10 cm?
Answer: Arc 15.7 + diameter 10 = 25.7 cm.
Did you get it right?
8Recap Recap
- Circumference: C = πd or C = 2πr
- π ≈ 3.14, the ratio circumference ÷ diameter
- Diameter = 2 × radius
- A semicircle's perimeter includes the diameter
- Circumference is a length in cm/m, not cm²