Lesson 12: Area of a Circle

§Text Passage

1Summary 重點摘要

Area of circle: A = π × r × r = πr². Don't confuse with circumference. Learn to calculate circle area and sector area.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Circle Area FormulaThe circle area formula is A = πr²: once the radius is known, the area can be worked out. Example: a circle of radius 5 cm has area 3.14 × 5² = 3.14 × 25 = 78.5 cm². Square the radius first, then multiply by 3.14; reversing the order gives a wrong answer. Do not work out 5² as 5 × 2 = 10; squaring means multiplying the number by itself, 5 × 5 = 25. Square first, then multiply by π — the order beginners slip on most often.
2. Block 2: Known DiameterIf the question gives the diameter, divide it by 2 first to get the radius, then substitute. Example: a circle of diameter 10 cm has radius 10 ÷ 2 = 5 cm, so the area = 3.14 × 5² = 78.5 cm². Squaring the diameter by mistake is the most common error, so check which one is given. Check with the relation between the two: a circle of radius 5 cm has a diameter of exactly 10 cm, matching the question, so the conversion is right before substituting.
3. Block 3: Area vs CircumferenceCircumference C = 2πr is a length in cm, while area A = πr² is a surface in cm²; never mix them. Read what the question asks: 'how long is the fence' or 'how far is one lap' means perimeter; 'how big is the grass' means area. Decide first, then choose the formula. Memory aid: perimeter asks how long one lap is (one-dimensional, cm), while area asks how much surface is covered (two-dimensional, cm²); a correct number with a wrong unit still loses marks.
4. Block 4: Semicircle AreaA semicircle's area is half of the full circle: semicircle area = πr² ÷ 2. Example: for radius 4 cm, area = 3.14 × 4² ÷ 2 = 3.14 × 16 ÷ 2 = 25.12 cm². No diameter is added here; adding it is only needed when finding the perimeter of a semicircle. Only when the question asks for the full perimeter of the semicircle must the arc and the diameter be added: for radius 4 cm, perimeter = 3.14 × 4 + 8 = 20.56 cm; finding area and finding perimeter are two different questions.
5. Block 5: Quarter Circle AreaA quarter circle is a 90° sector, and its area is one quarter of the full circle: area = πr² ÷ 4. Example: for radius 6 cm, quarter-circle area = 3.14 × 36 ÷ 4 = 28.26 cm². In the same way, three quarters of the circle is the full area times 34 — divide according to the share. You may also find the full circle first and divide by 4: 3.14 × 36 = 113.04, then 113.04 ÷ 4 = 28.26 cm², the same as writing 3.14 × 36 ÷ 4 in one line.
6. Block 6: Word Problem: GardenExample: a round flower bed has radius 2 m and grass is to be laid inside; how big is the grass? This is an area question: 3.14 × 2² = 3.14 × 4 = 12.56 m². Finish with a sentence such as 'the grass covers 12.56 square metres', writing m² because the question asks about size, not length. If the diameter 4 m is given, convert to the radius first: 4 ÷ 2 = 2 m, and the result is still 12.56 m²; word problems often state the diameter, so confirm which one is given.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Circle area = πr²
• Area is 2D, so r²
• Circumference is 1D, so 2πr
• Semicircle area = circle area ÷ 2
• Quarter circle area = circle area ÷ 4