Circles: circumference and area
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给教育者： 面向 Circles: circumference and area（KS3 Maths，Geometry & Measures）的即用型课程幻灯片, 复习笔记——用在你的课堂上，或将该知识点作为学习者可实时游玩的互动课堂活动来运行。
课程笔记
Parts of a Circle
- Centre: the point that is the same distance from every point on the circle.
- Radius (r): a line segment from the centre to any point on the circle; also its length.
- Diameter (d): a line segment through the centre with both endpoints on the circle; the longest chord, and d = 2r.
- Circumference: the distance all the way around the circle.
- Chord: a line segment with both endpoints on the circle; the diameter is the longest possible chord.
- Tangent: a straight line that touches the circle at exactly one point.
- Arc: any connected part of the circle.
- Sector: the region bounded by two radii and an arc; segment: the region bounded by a chord and an arc.
The Number π
- π (π) is the ratio of a circle's circumference to its diameter: π = C / d.
- This ratio is the same for every circle, whatever its size.
- π is approximately 3.14159… and is irrational, so it never ends or repeats.
- In calculations, use the π button on your calculator, or 3.14 as a rough approximation.
Circumference
- Circumference from diameter: C = πd.
- Circumference from radius: C = 2πr, because d = 2r.
- Example: a circle with r = 5 cm has C = 2 × π × 5 = 10π cm ≈ 31.4 cm.
- Leave answers in terms of π when asked for an exact value; otherwise round as instructed.
Area
- Area of a circle: A = πr².
- Square the radius first, then multiply by π.
- Example: a circle with r = 5 cm has A = π × 52 = 25π cm² ≈ 78.5 cm².
- If you are given the diameter, halve it to get the radius before using A = πr².
Semicircles and Quarter Circles
- A semicircle is half a circle: its arc length is half the circumference, C/2 = πr.
- Area of a semicircle: A = (πr²)/2.
- A quarter circle is one quarter of a circle: arc length = (2πr)/4 = (πr)/2.
- Area of a quarter circle: A = (πr²)/4.
- For a semicircle or quarter circle, remember to add the straight edges when finding the perimeter of the whole shape.
Arc Length and Sector Area
- An arc or sector is a fraction of the whole circle, set by the angle at the centre.
- Fraction of the circle = angle/360°.
- Arc length = (angle/360°) × πd = (angle/360°) × 2πr.
- Sector area = (angle/360°) × πr².
- Example: a sector with angle 90° is 90/360 = 1/4 of the circle, so its area is (πr²)/4.
Exact Answers and Rounding
- An answer in terms of π keeps the symbol π, e.g. 10π cm or 25π cm².
- A rounded answer replaces π with a decimal, e.g. 31.4 cm or 78.5 cm².
- Check whether the question asks for an exact answer or a decimal before you finish.
- Give units with every answer: cm, cm², m, m² and so on.
Key Facts to Remember
- d = 2r and r = d/2.
- C = πd = 2πr.
- A = πr².
- A circle is the boundary; the region inside it is called a disc.
- π is the same value for every circle, which is why these formulas work for all circles.
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练习题
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1.Which term means the distance around a circle?
Easy- ARadius
- BDiameter
- CCircumference
- DArea
2.Which formula gives the circumference of a circle with radius r?
Easy- AC = \π r2
- BC = 2\π r
- CC = \π d2
- DC = \frac{r}{2}
3.Which formula gives the area of a circle with radius r?
Easy- AA = 2\π r
- BA = \π d
- CA = \π r2
- DA = \frac{1}{2}\π r2
4.What is π defined as?
Medium- AThe ratio of area to radius
- BThe ratio of circumference to diameter
- CThe ratio of diameter to radius
- DThe ratio of circumference to radius
5.Which of the following are parts of a circle? (select all that apply)
Medium- AChord
- BTangent
- CSector
- DHypotenuse
- EDiameter
6.The diameter of a circle is twice the length of its radius.
EasyTrue or false?
7.A tangent is a straight line that crosses a circle at two points.
EasyTrue or false?
8.Match each circle term to its definition.
Medium- Radius
- Chord
- Tangent
- Arc
- A straight line touching the circle at exactly one point
- A line segment from the centre to a point on the circle
- A line segment joining two points on the circle
- A connected part of the circle's edge
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