Lesson 10: Introduction to Circles

§Text Passage

1Summary 重點摘要

A circle has centre, radius, diameter and circumference. Diameter = 2 × radius. Learn to draw circles and know the parts.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: CentreThe centre is the middle point of a circle, usually labelled O. When you draw a circle with a compass, the fixed needle point is the centre. Why does the centre matter? Because the centre decides where the circle sits on the paper but not how big it is: move the centre and the whole circle moves with it, unchanged in size, while the gap between the needle and the pencil is what sets the size. Every point on the circle is the same distance from the centre — this is the very definition of a circle and the reason a compass can draw one. For example, to draw a large circle on the playground, one person stands at the centre holding a taut rope while another walks round holding its end; the unchanging rope length traces the circle. A common mistake is to let the needle slip while drawing, shifting the centre so the line fails to close into a circle.
2. Block 2: RadiusA radius is a line segment from the centre to any point on the circle, usually written r. The longer the radius, the bigger the circle: a circle of radius 5 cm is larger than one of radius 3 cm. With a compass, the gap between the needle and the pencil is the radius. Why are all the radii of one circle equal? Because a circle is defined as the figure formed by all the points at one fixed distance from the centre; every radius is exactly that distance, so they must all be equal — measure ten radii of one circle and every length matches. Measure the radius with a ruler and open the compass accordingly before drawing. A common mistake is to think the radius runs through the centre to the far side, which is actually the diameter; a radius runs from the centre to the circle and is half the diameter.
3. Block 3: DiameterA diameter is a line segment passing through the centre with both ends on the circle, usually written d, and it is the longest segment in the circle. The relation is: diameter = 2 × radius. For example, a circle of radius 5 cm has diameter 2 × 5 = 10 cm; the other way round, a circle of diameter 14 cm has radius 14 ÷ 2 = 7 cm. Why is the diameter the longest? Because with both ends fixed on the circle, the segment reaches its greatest length only when it passes through the centre; every chord that misses the centre is shorter. Given either quantity, multiply or divide by 2 to find the other. A common mistake is to substitute the radius for the diameter and be out by a factor of two; when the question says radius or diameter, circle the word first, write the relation d = 2r, and check once before substituting.
4. Block 4: CircumferenceThe circumference is the length all the way around the circle, its boundary. Why must it not be confused with area? Because the circumference is the length of a curved line, measured in cm, while the area is the size of the flat surface inside, measured in cm² — different meanings and different units. For example, walking once around a round flower bed covers the circumference, while the size of the soil inside is the area; a lace trim around the edge of a round table needs the circumference, and a sheet of glass over the tabletop needs the area. To measure a circumference, wrap a tape around the circle and read it directly, or wrap a string once round, straighten it, and measure with a ruler. A common mistake is to swap the units, writing cm² for a circumference or cm for an area; decide first whether you are finding a length or a size, then choose the unit.
5. Block 5: Drawing a CircleDrawing a circle with a compass has three steps: first, open the compass to the radius you need and measure it with a ruler; second, fix the needle point at the position of the centre; third, hold the top and turn one full round, and the pencil draws the circle. Why must the needle stay still? Because the needle is the centre; once it moves, the distances are no longer equal, the line fails to close, and the curve is not a circle. For example, to draw a circle of radius 4 cm: open the two legs to 4 cm, press the needle gently on the paper at the chosen point, hold the top handle, and sweep one full turn. A common mistake is to grip a leg of the compass while turning, letting the angle between the legs change so the radius drifts; hold only the top handle and keep the opening constant from start to finish.
6. Block 6: SymmetryThe circle is the shape with the most lines of symmetry — infinitely many: every straight line through the centre is an axis of symmetry, and folding along it makes the two halves cover each other exactly. Why so many? Because every point on the circle is equidistant from the centre, the figure looks the same in every direction, and a fold along any line through the centre always matches the two sides perfectly. Compare other figures: a square has only 4 lines of symmetry and a general rectangle only 2. For example, fold a paper circle along a diameter and the two halves coincide; turn the circle through any angle and it looks unchanged. A common mistake is to think a circle has only the vertical and the horizontal axes, although every slanting diameter is one too; they cannot be counted not because there are few, but because there are infinitely many.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Centre is the middle
• Radius is centre to edge
• Diameter = 2 × radius
• Circumference is the perimeter
• Circle has infinitely many symmetry axes