Circles Arcs And Sectors

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Notes

Circle Basics

A **circle** is all points equidistant from a single centre.
**Circumference** is the perimeter of a circle.
**Diameter** $(d) = 2 \times$ **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: $r^{2}$ .
Multiply by $\pi : A = \pi r^{2}$ .
Area is measured in square units $(e.g., cm^{2}, m^{2})$ .
Example: radius 5.1 cm → area $= \pi \times 5.1^{2} = 81.7 cm^{2} (3 s.f.)$ .

Circumference of a Circle

Identify the diameter (double the radius).
Multiply diameter by $\pi : C = \pi d$ .
Or use $C = 2\pi r$ .
Circumference is measured in units of length (e.g., cm, m).
Example: radius 8 cm → circumference $= 2\pi \times 8 = 16\pi$ 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) $\times 2\pi r$ .
Step 1: Divide angle by 360.
Step 2: Find full circumference (2πr).
Step 3: Multiply fraction by circumference.
Example: θ=72°, $r=5$ cm → $arc = (\frac{72}{360})\times 2\pi \times 5 = 2\pi$ cm.

Area of a Sector

Area of sector = (θ/360) $\times \pi r^{2}$ .
Step 1: Divide angle by 360.
Step 2: Find full circle area $(\pi r^{2})$ .
Step 3: Multiply fraction by area.
Example: θ=72°, $r=5$ cm → area $= (\frac{72}{360})\times \pi \times 5^{2} = 15.7 cm^{2} (3 s.f.)$ .

Perimeter of a Sector

Perimeter $= arc$ length $+ 2 \times$ radius.
Includes the two straight radii and the curved arc.
Example: radius 8 cm, angle 165° → $arc = (\frac{165}{360})\times 2\pi \times 8$ ≈ 23.0 cm, perimeter ≈ $23.0 + 16 = 39.0$ cm.

Working with Semicircles and Quarter Circles

Semicircle: angle $= 180^{\circ}$ , so area = ½πr², arc length $= \pi r$ .
Quarter circle: angle $= 90^{\circ}$ , so area = ¼πr², arc length = ½πr.
Perimeter of semicircle $= \pi r + 2r (arc + diameter)$ .
Example: diameter 16 cm → radius 8 cm, area $= 32\pi cm^{2}$ , perimeter $= (8\pi +16)$ 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.

Circle with radius and diameter

Sector with angle θ

Major and minor arcs

Semicircle with diameter

Practice questions

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1.What is the formula for the area of a circle with radius r?
Easy
- A $A = \pi r^{2}$
- B $A = 2\pi r$
- C $A = \pi d$
- D $A = \pi r$
2.What is the circumference of a circle with diameter d?
Easy
- A $C = \pi d$
- B $C = 2\pi d$
- C $C = \pi r$
- D $C = \pi r^{2}$
3.A circle has radius 5 cm. What is its area?
Easy
- A $25\pi cm^{2}$
- B $10\pi cm^{2}$
- C $5\pi cm^{2}$
- D $\pi cm^{2}$
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
- A $6\pi cm^{2}$
- B $12\pi cm^{2}$
- C $36\pi cm^{2}$
- D $\pi cm^{2}$
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. $(Use \pi = \frac{22}{7})$
Medium
- A44 cm
- B22 cm
- C154 cm
- D88 cm
8.A circle has circumference 12π cm. What is its radius?
Medium
- A6 cm
- B12 cm
- C24 cm
- D3 cm

Summary

## 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 $\theta = 360 - \text{minor angle}$.

Key terms

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.

Slides

Area & Circumference of Circles

A circle's circumference (perimeter) is $C = \pi d$ or $C = 2\pi r$ .
The area of a circle is $A = \pi r^{2}$ .
π (π) is approximately 3.14159.

What is the circumference of a circle with radius 5 cm?

5π cm
10π cm
25π cm
2π cm

Arc Length & Sector Area

An arc is part of the circumference; a sector is the region between two radii and an arc.
Arc length = (θ/360) $\times 2\pi r$ .
Sector area = (θ/360) $\times \pi r^{2}$ .
θ is the angle at the centre in degrees.

The formula for the area of a sector is $(____/360) \times \pi r^{2}$ .

Finding Arc Length

Step 1: Write the fraction θ/360.
Step 2: Find the full circumference (2πr).
Step 3: Multiply the fraction by the circumference.

A sector has radius 6 cm and angle 60°. What is the arc length?

π cm
2π cm
6π cm
12π cm

Finding Sector Area

Step 1: Write the fraction θ/360.
Step 2: Find the full area $(\pi r^{2})$ .
Step 3: Multiply the fraction by the area.

A sector of radius 10 cm and angle 72° has area $____\pi cm^{2}$ .

Perimeter of a Sector

The perimeter of a sector $= arc$ length $+ 2 \times$ radius.
Remember to include both straight edges (radii).

A sector has radius 8 cm and arc length 4π cm. What is its perimeter?

$4\pi + 8$ cm
$4\pi + 16$ cm
$8\pi + 16$ cm
$8\pi + 8$ cm

Working Backwards

If given arc length or sector area, you can find θ or r.
Rearrange the formula: θ $= (arc$ length $\times 360) / (2\pi r)$ .
Or $r = √( (sector$ area $\times 360) /$ (θπ) ).

Which of these are true?

Arc length = (θ/360) $\times \pi r^{2}$
Sector area = (θ/360) $\times \pi r^{2}$
Perimeter of sector $= arc$ length + 2r

Exact Answers in Terms of π

Sometimes you must leave π in the answer $(e.g., 32\pi cm^{2})$ .
Do not multiply by π on your calculator; keep it symbolic.

A semicircle has radius 4 cm. Its area $is ____\pi cm^{2}$ .

Common Exam Tips

Area is in square units $(cm^{2}, m^{2})$ .
Lengths are in units (cm, m).
Always check if the angle is in degrees.

A sector has area $30\pi cm^{2}$ and radius 6 cm. What is the angle θ?

150°
300°
60°
90°

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