Circle Properties

1Summary 重點摘要

Radius, diameter, circumference, pi — basics of circle.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. Radius and diameterThe radius r is the distance from the centre O to any point on the circle; the diameter d is a segment through the centre with both ends on the circle. They are linked by d = 2r — the diameter is always twice the radius. For example, a circle of radius 3 cm has diameter 6 cm. A common error is halving or doubling the wrong one, so check whether the question gives r or d before you start.
2. 2. CircumferenceThe distance around a circle is its circumference, written in two equivalent ways: circumference = πd = 2πr. A circle of diameter 10 cm has circumference π × 10 ≈ 31.4 cm; using the radius gives 2 × π × 5 ≈ 31.4 cm, the same answer. Circumference is a length measured in cm — do not confuse it with the area formula πr², which differs by one square and is a classic exam trap.
3. 3. Area of a circleThe area of a circle is πr²: square the radius first, then multiply by π. For radius 3 cm, area = π × 3² = π × 9 ≈ 28.3 cm². The usual mistakes are writing π × 3 × 2 or forgetting the square. Another trap: when the diameter is given, say d = 10, halve it first to get r = 5, then compute π × 5² = 78.5 cm². Area units are square units (cm²) — never omit them.
4. 4. Getting to know piPi (π) is defined as the circumference of any circle divided by its diameter, and this ratio is the same for every circle. Its value is about 3.14159265..., an infinite non-repeating decimal. Exams usually allow 3.14 or the fraction 227; these differ slightly (about 3.1400 versus 3.1429), so use whichever the question specifies or your answer drifts a little. π is a Greek letter — do not write it as the English letter n.
5. 5. Chord and arcA chord is a straight segment joining two points on the circle; an arc is a curved part of the circle itself. The diameter is the chord passing through the centre, and it is the longest chord of all, splitting the circle into two semicircles. To classify a segment, first check that both ends lie on the circle (a chord), then whether it passes through the centre (a diameter). These terms underpin later work, so keep the definitions exact.
6. 6. Finding the radius backwardsTo find the radius from a given circumference, rearrange the formula: since circumference = 2πr, the radius is circumference ÷ (2π). For example, with circumference 31.4 cm and π ≈ 3.14, r = 31.4 ÷ (2 × 3.14) = 31.4 ÷ 6.28 = 5 cm. Similarly, from an area use r = √(area ÷ π). Such reverse questions are common in exams: write the original formula first and rearrange step by step instead of jumping mentally, which is where direction errors creep in.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• d = 2r
• Circumference = 2πr = πd
• Area = πr²
• π ≈ 3.14
• Longest chord is diameter