Circle area (intro)

1Summary 重點摘要

Area = π × r × r.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The formulaCircle area = π × radius × radius, written A = πr². The formula needs only the radius: square it first, then multiply by π, in that order. Why only the radius? Because the size of a circle is fixed entirely by its radius; once the radius is set, the circle is set. For a radius of 2 cm, first work out 2 × 2 = 4, then 3.14 × 4 = 12.56 cm². A common mistake is multiplying by the radius only once, writing 3.14 × 2 = 6.28, which is exactly half the true area; keep in mind that the radius must be squared, the first rule of circle area.
2. Step 2: What r² meansr² means the radius multiplied by itself: for r = 2, r² = 2 × 2 = 4. It is not r × 2, and not 2r; with r = 3, r² = 9 while 2r = 6, and the difference appears at once. Why does the area formula square the radius? Because area is fundamentally the product of two lengths, so the radius, being a length, must multiply itself to match a surface. Seeing a square sign should always suggest self-multiplication. A common error is computing r² as r × 2, such as writing 5² as 10 instead of 25; settle the square first, since this is the most typical slip in circle-area problems.
3. Step 3: SubstituteFor a circle of radius 2 cm: A = 3.14 × 2 × 2 = 3.14 × 4 = 12.56 cm². The calculation has two steps: square the radius to get 4, then multiply by π to get 12.56, and write both steps clearly for easy checking. Why write in steps? Because finishing in one breath hides errors, and a mistake then cannot be located; separate writing lets the squaring and the multiplying be checked one at a time. With larger numbers, for radius 10, first 10 × 10 = 100, then 3.14 × 100 = 314 cm². A common fault is omitting the square unit cm²; an area answer is complete only with square units.
4. Step 4: Compare with circumferenceFor radius 2 the circumference is 2 × 3.14 × 2 = 12.56 cm, coincidentally the same number as the area, but it is only a coincidence, not a rule. Why compare them? Because circumference and area are most easily confused: the circumference is the length of a curved line, measured in cm, while the area is the size of a surface, measured in cm². Equal numbers, completely different meanings; with radius 3 the circumference 18.84 and the area 28.26 no longer match. A common error is using the area formula for a perimeter question, or swapping the units; read the question first to see whether it asks for the circumference or the area.
5. Step 5: Starting from the diameterWhen the question gives the diameter, halve it to find the radius before using the formula: for d = 10 cm, r = 5 cm, so A = 3.14 × 5 × 5 = 3.14 × 25 = 78.5 cm². Why convert first? Because the formula uses the radius only, and substituting the diameter as if it were the radius makes the area 4 times too large: using 10 gives 314, exactly 4 times the correct 78.5. The common error is forgetting to halve, squaring the diameter straight away; circle the word diameter in the question as a reminder, find the radius first, and the trap is avoided.
6. Step 6: The semicircleA semicircle's area is half the full circle: with radius 2 cm the circle's area is 12.56 cm², so the semicircle is 12.56 ÷ 2 = 6.28 cm². Working out the whole circle and then halving is simpler than memorising another formula and avoids mixing up the coefficient. Why exactly half? Because the diameter cuts the circle into two completely equal parts, identical in shape and size. Whenever a question mentions a semicircle, compute the full circle first and divide by 2 at the end. A common error is forgetting to halve and giving the full circle as the semicircle's area; watch for the word half in the question.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Circle area = π × r × r (πr²)
• r² squares the radius — not r × 2
• Halve the diameter before using the formula
• Semicircle area = circle area ÷ 2
• Area uses cm²; circumference uses cm