Circles Arcs And Sectors

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Apuntes de la lección

Circle Basics

A circle is all points equidistant from a single centre.
Circumference is the perimeter of a circle.
Diameter (d) = 2 × radius (r).
π (π) ≈ 3.14159 is the ratio of circumference to diameter.
Area formula: A = πr².
Circumference formulas: C = πd or C = 2πr.
Answers may be left 'in terms of π' for exact value.

Area of a Circle

Identify the radius (half the diameter).
Square the radius: r².
Multiply by π: A = πr².
Area is measured in square units (e.g., cm², m²).
Example: radius 5.1 cm → area = π × 5.1² = 81.7 cm² (3 s.f.).

Circumference of a Circle

Identify the diameter (double the radius).
Multiply diameter by π: C = πd.
Or use C = 2πr.
Circumference is measured in units of length (e.g., cm, m).
Example: radius 8 cm → circumference = 2π×8 = 16π cm.

Arcs and Sectors

An arc is part of the circumference; a sector is a 'pizza slice' enclosed by two radii and an arc.
Two arcs/sectors exist: minor (smaller) and major (larger).
The angle at the centre is often labelled θ (θ).
Fraction of full circle = θ/360.

Arc Length

Arc length = (θ/360) × 2πr.
Step 1: Divide angle by 360.
Step 2: Find full circumference (2πr).
Step 3: Multiply fraction by circumference.
Example: θ=72°, r=5 cm → arc = (72/360)×2π×5 = 2π cm.

Area of a Sector

Area of sector = (θ/360) × πr².
Step 1: Divide angle by 360.
Step 2: Find full circle area (πr²).
Step 3: Multiply fraction by area.
Example: θ=72°, r=5 cm → area = (72/360)×π×5² = 15.7 cm² (3 s.f.).

Perimeter of a Sector

Perimeter = arc length + 2 × radius.
Includes the two straight radii and the curved arc.
Example: radius 8 cm, angle 165° → arc = (165/360)×2π×8 ≈ 23.0 cm, perimeter ≈ 23.0 + 16 = 39.0 cm.

Working with Semicircles and Quarter Circles

Semicircle: angle = 180°, so area = ½πr², arc length = πr.
Quarter circle: angle = 90°, so area = ¼πr², arc length = ½πr.
Perimeter of semicircle = πr + 2r (arc + diameter).
Example: diameter 16 cm → radius 8 cm, area = 32π cm², perimeter = (8π+16) cm.

Problem Solving Tips

Always write down given values (r, θ, arc length, area).
Pick correct formula and substitute carefully.
Use calculator for decimal answers; round to required significant figures.
For 'in terms of π', do not multiply by π.
Check units: area in square units, length in units.

Diapositivas

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Preguntas de práctica

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1.What is the formula for the area of a circle with radius r?
Easy
- AA = πr²
- BA = 2πr
- CA = πd
- DA = πr
2.What is the circumference of a circle with diameter d?
Easy
- AC = πd
- BC = 2πd
- CC = πr
- DC = πr²
3.A circle has radius 5 cm. What is its area?
Easy
- A25π cm²
- B10π cm²
- C5π cm²
- Dπ cm²
4.A circle has diameter 10 cm. What is its circumference?
Easy
- A10π cm
- B20π cm
- C5π cm
- D100π cm
5.A sector of a circle has radius 6 cm and angle 60°. What is the area of the sector?
Medium
- A6π cm²
- B12π cm²
- C36π cm²
- Dπ cm²
6.A sector has radius 4 cm and angle 90°. What is the arc length?
Medium
- A2π cm
- B4π cm
- Cπ cm
- D8π cm
7.The radius of a circle is 7 cm. Calculate the circumference. (Use π = 22/7)
Easy
- A44 cm
- B22 cm
- C154 cm
- D88 cm
8.A circle has circumference 12π cm. What is its radius?
Easy
- A6 cm
- B12 cm
- C24 cm
- D3 cm

Más preguntas

9.A sector has area 15π cm² and radius 5 cm. What is the sector angle?
Medium
- A216°
- B120°
- C60°
- D72°
10.An arc of length 4π cm comes from a circle of radius 6 cm. What is the angle subtended by the arc?
Medium
- A120°
- B60°
- C90°
- D180°
11.What is the diameter of a circle with radius 3.5 cm?
Easy
- A7 cm
- B1.75 cm
- C3.5 cm
- D10.5 cm
12.A semicircle has diameter 10 cm. What is its perimeter?
Medium
- A(5π + 10) cm
- B10π cm
- C5π cm
- D(10π + 10) cm
13.A sector has radius 8 cm and arc length 6.4 cm. What is the area of the sector?
Hard
- A25.6 cm²
- B51.2 cm²
- C12.8 cm²
- D20.8 cm²
14.What is the value of π approximately?
Easy
- A3.14
- B2.14
- C1.57
- D4.14
15.A circle has area 49π cm². What is its radius?
Easy
- A7 cm
- B14 cm
- C49 cm
- D3.5 cm
16.A sector has perimeter 20 cm and radius 6 cm. What is the arc length?
Medium
- A8 cm
- B12 cm
- C14 cm
- D6 cm
17.A circle has radius 3 cm. What is its circumference?
Easy
- A6π cm
- B9π cm
- C3π cm
- D12π cm
18.A circle has diameter 8 cm. What is its area?
Easy
- A16π cm²
- B64π cm²
- C8π cm²
- D32π cm²
19.A sector of a circle has radius 10 cm and angle 36°. What is the area of the sector?
Medium
- A10π cm²
- B36π cm²
- C100π cm²
- D3.6π cm²
20.A sector has radius 7 cm and arc length 14π/3 cm. What is the sector angle?
Hard
- A120°
- B60°
- C90°
- D150°
21.A circle has area 36π cm². What is its circumference?
Medium
- A12π cm
- B6π cm
- C18π cm
- D24π cm
22.A sector has perimeter 25 cm and radius 7 cm. What is the arc length?
Medium
- A11 cm
- B18 cm
- C14 cm
- D7 cm
23.The arc of a sector is 8π cm and the radius is 12 cm. What is the sector angle?
Medium
- A120°
- B60°
- C90°
- D150°
24.What is the area of a semicircle with diameter 10 cm?
Medium
- A12.5π cm²
- B25π cm²
- C50π cm²
- D10π cm²
25.A sector has area 30π cm² and radius 6 cm. What is the sector angle?
Medium
- A300°
- B150°
- C60°
- D120°
26.A circle has circumference 30π cm. What is the area of a sector with angle 72°?
Medium
- A45π cm²
- B90π cm²
- C30π cm²
- D60π cm²
27.A circle has radius 8 cm. What is the length of an arc subtended by an angle of 135°?
Medium
- A6π cm
- B12π cm
- C3π cm
- D9π cm
28.What is the formula for the area of a sector with radius r and angle θ?
Easy
- A(θ/360) × πr²
- B(θ/360) × 2πr
- Cπr²
- D2πr
29.A semicircle has radius 6 cm. What is its perimeter?
Medium
- A6π + 12 cm
- B12π cm
- C6π cm
- D12π + 12 cm
30.The area of a sector is 40 cm² and the radius is 8 cm. What is the arc length?
Hard
- A10 cm
- B5 cm
- C20 cm
- D16 cm
31.What is the circumference of a circle with radius 2.5 cm?
Easy
- A5π cm
- B2.5π cm
- C6.25π cm
- D10π cm
32.A sector has arc length 5π cm and radius 9 cm. What is the sector angle?
Medium
- A100°
- B50°
- C200°
- D150°
33.What is the formula for the circumference of a circle with radius r?
Easy
- AC = πr
- BC = 2πr
- CC = πr²
- DC = 2πr²
34.The circumference of a circle is 31.4 cm. What is its radius? (Use π = 3.14)
Easy
- A5 cm
- B10 cm
- C2.5 cm
- D15.7 cm
35.The area of a circle is 78.5 cm². What is its radius? (Use π = 3.14)
Medium
- A5 cm
- B10 cm
- C25 cm
- D2.5 cm
36.A sector has radius 6 cm and arc length 4π cm. What is the sector angle?
Medium
- A120°
- B60°
- C90°
- D30°
37.The diameter of a circle is twice its radius.
Easy
True or false?
38.A semicircle has diameter 14 cm. What is its perimeter? (Use π = 22/7)
Medium
- A36 cm
- B22 cm
- C44 cm
- D58 cm
39.Match each formula to its correct description.
Easy
- Area of a circle
- Circumference of a circle
- Area of a sector
- Arc length
- πr²
- 2πr
- θ/360 × πr²
- θ/360 × 2πr
40.Arrange the following steps in the correct order to find the area of a sector with radius 5 cm and angle 72°.
Medium
- Multiply by π × 5²
- Divide the angle by 360
- Calculate 72/360 × π × 25
- Write down the formula: area = θ/360 × πr²
41.A circle has radius 10 cm. Which of the following statements are correct? (Select all that apply.)
Medium
- AIts diameter is 20 cm
- BIts circumference is 20π cm
- CIts area is 100π cm²
- DIts area is 100 cm²
- EIts circumference is 10π cm
42.The arc length of a sector with angle 90° and radius 6 cm is 3π cm.
Medium
True or false?
43.A circle has radius 4.5 cm. Calculate its circumference. (Use π = 3.142)
Easy
44.A circle has diameter 12 cm. Find its area in terms of π.
Easy
45.A sector has radius 9 cm and angle 40°. Calculate the area of the sector. (Use π = 3.142)
Medium
46.A sector has area 50π cm² and radius 10 cm. Find the arc length of this sector.
Hard
47.Complete the sentence.
Easy
The perimeter of a circle is called its ____.
48.A quarter-circle has radius 8 cm. Find its area in terms of π.
Medium
49.A sector of a circle has area 20π cm² and arc length 5π cm. Find the radius of the sector.
Hard
50.Complete the sentence about a semicircle.
Medium
A semicircle has diameter 14 cm. Its perimeter is ____ cm (use π = 22/7).
51.A sector has radius 12 cm and arc length 8π cm. Calculate the angle of the sector.
Medium
52.A circle has radius 4.5 cm. Calculate its circumference. (Use π = 3.142)
Easy
- A33.934 cm
- B28.278 cm
- C31.106 cm
- D25.45 cm
53.A circle has diameter 12 cm. Find its area in terms of π.
Easy
- A32 π cm²
- B40 π cm²
- C44 π cm²
- D36 π cm²
54.A sector has radius 9 cm and angle 40°. Calculate the area of the sector. (Use π = 3.142)
Medium
- A25.45 cm²
- B28.278 cm²
- C31.106 cm²
- D33.934 cm²
55.A sector has area 50π cm² and radius 10 cm. Find the arc length of this sector.
Hard
- A9 π cm
- B12 π cm
- C11 π cm
- D10 π cm
56.A quarter-circle has radius 8 cm. Find its area in terms of π.
Medium
- A17 π cm²
- B18 π cm²
- C16 π cm²
- D15 π cm²
57.A sector of a circle has area 20π cm² and arc length 5π cm. Find the radius of the sector.
Hard
- A7 cm
- B9 cm
- C8 cm
- D10 cm

Apuntes de repaso

## Circles, Arcs & Sectors - Quick Revision ### Key Formulas - **Circumference**: \(C = \pi d\) or \(C = 2\pi r\) - **Area of a circle**: \(A = \pi r^2\) - **Arc length**: \(\text{Arc length} = \frac{\theta}{360} \times 2\pi r\) - **Area of a sector**: \(\text{Area} = \frac{\theta}{360} \times \pi r^2\) ### Important Notes - \(\pi\) (pi) ≈ 3.14159. You may be asked to give answers **in terms of \(\pi\)** or as an exact value. - The fraction \(\frac{\theta}{360}\) represents the proportion of the full circle. - **Perimeter of a sector** = arc length + 2 × radius. - For semicircles: area = \(\frac{1}{2}\pi r^2\), curved length = \(\pi r\), perimeter = \(\pi r + 2r\). ### Common Problem Types - Given radius and angle, find arc length or sector area. - Given arc length and radius, find angle (rearrange formula). - Given sector area and radius, find angle. - Find perimeter of a sector (remember to add both radii). - Work with compound shapes (e.g., rectangle + semicircles). - Convert units (e.g., inches to cm) when needed. ### Examiner Tips - Always check if the answer should be in exact form (in terms of \(\pi\)) or rounded to a given number of significant figures. - Area is in square units, length in units. - For major arcs/sectors, use \(\theta = 360 - \text{minor angle}\).

Circumference: The perimeter of a circle; formula: \(C = \pi d\) or \(C = 2\pi r\).
Arc: A part of the circumference of a circle. The smaller arc is the minor arc, the larger is the major arc.
Sector: The region enclosed by two radii and an arc. The smaller is the minor sector, the larger is the major sector.
Arc length: The length of an arc; formula: \(\text{Arc length} = \frac{\theta}{360} \times 2\pi r\), where \(\theta\) is the angle at the centre in degrees.
Area of a sector: The area of a sector; formula: \(\text{Area} = \frac{\theta}{360} \times \pi r^2\).
π (pi): The ratio of a circle's circumference to its diameter, approximately 3.14159.

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