Circles Arcs And Sectors

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Lesson notes

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.

Slides

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Practice questions

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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

Revision notes

## 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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