Circles Arcs And Sectors

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課程筆記

Circle Basics

A circle is all points equidistant from a single centre point.
The circumference is the perimeter of the circle.
The diameter (d) is twice the radius (r): d = 2r.
π (π) ≈ 3.14159 is the ratio of circumference to diameter.
Answers may be given as 'exact value' or 'in terms of π'.

Area of a Circle

Formula: Area = πr² (given in exam).
Identify the radius (half the diameter).
Square the radius, then multiply by π.
Area is measured in square units (cm², m², etc.).

Circumference of a Circle

Formulae: C = πd or C = 2πr (given in exam).
Identify the diameter (or radius).
Multiply the diameter by π (or 2πr).
Circumference is measured in units of length (cm, m, etc.).

Arcs

An arc is part of the circumference between two points.
The minor arc is the smaller arc; the major arc is the larger.
Arc length formula: Arc length = (θ/360) × 2πr.
θ is the angle at the centre in degrees.

Sectors

A sector is the region enclosed by two radii and an arc.
The minor sector is the smaller sector; the major sector is the larger.
Sector area formula: Area = (θ/360) × πr².
A sector looks like a pizza slice.

Working with Formulae

Write down known values (r, d, θ).
Pick the correct formula (area, circumference, arc length, sector area).
Substitute values and solve.
For arcs and sectors, first find the fraction θ/360 of the full circle.

Semicircles

A semicircle is half a circle (θ = 180°).
Area of semicircle = (1/2)πr².
Perimeter of semicircle = (1/2)×2πr + d = πr + 2r.
Remember to include the diameter (straight edge) in perimeter.

Common Exam Tips

Area is always in square units; circumference/length in linear units.
Leave answers in terms of π if asked for 'exact value'.
For compound shapes (e.g., track), break into circles/rectangles.
Check whether to use radius or diameter in each formula.

投影片

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練習題

免費預覽——55 題中的 8 題。註冊即可查看全部。

1.A circle has diameter 6 cm. Calculate the area of the circle. (Use π = 3.14)
Easy
- A28.26 cm²
- B18.84 cm²
- C113.04 cm²
- D9.42 cm²
2.Calculate the circumference of a circle with radius 4.5 cm. (Use π = 3.14)
Easy
- A28.26 cm
- B14.13 cm
- C9.42 cm
- D63.585 cm
3.A circle has diameter 7 cm. Show that the circumference is 21.99 cm correct to 2 decimal places. Which calculation gives this value?
Easy
- Aπ × 7
- B2 × π × 7
- Cπ × 3.5
- D2 × π × 3.5
4.A circle has a circumference of 56 mm. Work out the radius of this circle. (Use π = 3.14)
Medium
- A8.92 mm
- B17.83 mm
- C28 mm
- D8.75 mm
5.Rachel has a pond in the shape of a circle. The circumference is 4.25 m. Calculate the diameter in centimetres. (Use π = 3.14)
Medium
- A135.4 cm
- B67.7 cm
- C13.54 cm
- D1.354 cm
6.The diagram shows a semicircle with diameter 9 cm. Calculate the total perimeter of this semicircle. (Use π = 3.14)
Medium
- A23.13 cm
- B14.13 cm
- C28.26 cm
- D18.13 cm
7.A circular pool has a diameter of 8 m. Calculate the circumference of the pool. (Use π = 3.14)
Easy
- A25.12 m
- B50.24 m
- C12.56 m
- D200.96 m
8.A circular pool has a diameter of 8 m. Calculate the area of the pool. (Use π = 3.14)
Easy
- A50.24 m²
- B25.12 m²
- C200.96 m²
- D12.56 m²

複習筆記

## Circles, Arcs & Sectors ### Key Formulas - **Area of a circle**: \(A = \pi r^2\) - **Circumference**: \(C = \pi d\) or \(C = 2\pi r\) - **Arc length**: \(\frac{\theta}{360} \times 2\pi r\) - **Area of a sector**: \(\frac{\theta}{360} \times \pi r^2\) ### Definitions - **Radius** (\(r\)): distance from centre to circumference. - **Diameter** (\(d\)): \(d = 2r\). - **Circumference**: perimeter of a circle. - **Arc**: part of the circumference (minor/major). - **Sector**: region between two radii and an arc (minor/major). - **\(\pi\)**: constant ≈ 3.14159; ratio of circumference to diameter. ### Tips - Area is in square units (e.g., cm²), circumference in linear units (e.g., cm). - For semicircles: area = \(\frac{1}{2}\pi r^2\), perimeter = \(\pi r + 2r\). - To find radius from circumference: \(r = \frac{C}{2\pi}\). - To find diameter from circumference: \(d = \frac{C}{\pi}\). - For arcs/sectors: fraction = \(\frac{\theta}{360}\). - "In terms of \(\pi\)" means leave \(\pi\) in the answer (do not multiply). - Always check units (e.g., convert cm to m if needed). ### Common Question Types - Calculate area/circumference given radius or diameter. - Find radius/diameter given area or circumference. - Find arc length or sector area given angle and radius. - Perimeter of compound shapes (e.g., semicircle + diameter). - Shaded area problems (subtract areas).

Circle: Set of points equidistant from a centre.
Radius: Distance from centre to circumference; half the diameter.
Diameter: Distance across a circle through centre; twice the radius.
Circumference: Perimeter of a circle; \(C = \pi d\).
Pi (π): Constant ≈ 3.14159; ratio of circumference to diameter.
Arc: Part of the circumference; minor or major.
Sector: Region between two radii and an arc; minor or major.
Arc length: Length of an arc; \(\frac{\theta}{360} \times 2\pi r\).
Area of a sector: Area of a sector; \(\frac{\theta}{360} \times \pi r^2\).

字卡

What is the formula for the area of a circle?

A = πr²

What is the formula for the circumference of a circle?

C = πd or C = 2πr

How do you find the radius from the diameter?

Radius = diameter ÷ 2

How do you find the diameter from the radius?

Diameter = 2 × radius

A circle has diameter 6 cm. Calculate its area.

Area = π × (3)² = 9π cm² (or 28.27 cm²)

Calculate the circumference of a circle with radius 4.5 cm.

C = 2π × 4.5 = 9π cm (or 28.27 cm)

A circle has diameter 7 cm. Show that its circumference is 21.99 cm correct to 2 decimal places.

C = πd = π × 7 = 21.9911... ≈ 21.99 cm

A circle has circumference 56 mm. Find its radius.

r = C ÷ (2π) = 56 ÷ (2π) = 28/π ≈ 8.91 mm

The circumference of a pond is 4.25 m. Calculate its diameter in centimetres.

d = C ÷ π = 4.25 ÷ π ≈ 1.3528 m = 135.28 cm

What is the formula for the area of a sector?

Area = (θ/360) × πr²

What is the formula for the arc length of a sector?

Arc length = (θ/360) × 2πr

OAB is a sector with radius 9 cm and angle 30°. Calculate its area in terms of π.

Area = (30/360) × π × 9² = (1/12) × 81π = (27/4)π cm²

A semicircle has diameter 9 cm. Calculate its total perimeter.

Perimeter = (π × 9)/2 + 9 = 4.5π + 9 cm (or 23.14 cm)

A circular pool has diameter 8 m. Find its circumference and area.

Circumference = π × 8 = 8π m (≈ 25.13 m); Area = π × 4² = 16π m² (≈ 50.27 m²)

A circle has radius 11 cm. Find its circumference in terms of π.

C = 2π × 11 = 22π cm

In a circle of radius 11 cm, minor arc FH subtends angle 140° at centre. Calculate the length of minor arc FH.

Arc length = (140/360) × 2π × 11 = (7/18) × 22π = (77/9)π cm (≈ 26.88 cm)

A square is inscribed in a circle of radius 6 cm. Find the shaded area (area of circle minus area of square).

Area of circle = 36π cm²; side of square = 6√2 cm, area = 72 cm²; shaded area = 36π - 72 cm² (≈ 41.10 cm²)

Points P, Q, R on a semicircle of radius 8 cm, centre O, with angle POQ = 90°. Calculate the shaded area.

Shaded area = area of sector POQ - area of triangle POQ = (90/360)×π×8² - (1/2)×8×8 = 16π - 32 cm² (≈ 18.27 cm²)

A running track consists of a rectangle 120m by 60m with two semicircles on the shorter sides. Find the total length of the track.

Length = 2 × 120 + π × 60 = 240 + 60π m (≈ 428.50 m)

Shreva walks at 1.4 m/s around a track of length 428.50 m. How long does it take in minutes and seconds?

Time = 428.50 ÷ 1.4 ≈ 306.07 s = 5 minutes 6 seconds

A shape is made of two semicircles with same centre: large radius 7 cm, small radius 3 cm. Calculate the area of the shape.

Area = (1/2)π×7² - (1/2)π×3² = (1/2)π(49-9) = 20π cm² (≈ 62.83 cm²)

The area of a circle is 76.4 cm². Find its radius.

r = √(76.4/π) ≈ √24.32 ≈ 4.93 cm

A circle has diameter 38 mm. Calculate its circumference in centimetres.

C = π × 38 mm = 38π mm ≈ 119.38 mm = 11.938 cm

What is π (π)?

π is the ratio of a circle's circumference to its diameter, approximately 3.14159.

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