Circles Arcs And Sectors

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Notes

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 $(cm^{2}, m^{2}, etc.)$ .

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 $= 180^{\circ})$ .
Area of semicircle $= (\frac{1}{2})\pi r^{2}$ .
Perimeter of semicircle $= (\frac{1}{2})\times 2\pi r + d = \pi r + 2r$ .
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.

Circle with radius and diameter

Sector and arc

Semicircle with diameter

Major and minor arcs/sectors

Practice questions

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1.A circle has diameter 6 cm. Calculate the area of the circle. $(Use \pi = 3.14)$
Easy
- A $28.26 cm^{2}$
- B $18.84 cm^{2}$
- C $113.04 cm^{2}$
- D $9.42 cm^{2}$
2.Calculate the circumference of a circle with radius 4.5 cm. $(Use \pi = 3.14)$
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
- A $\pi \times 7$
- B $2 \times \pi \times 7$
- C $\pi \times 3.5$
- D $2 \times \pi \times 3.5$
4.A circle has a circumference of 56 mm. Work out the radius of this circle. $(Use \pi = 3.14)$
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. $(Use \pi = 3.14)$
Medium
- A135.4 cm
- B67.7 cm
- C13.54 cm
- D1.354 cm
6.OAB is a sector of a circle with radius 9 cm and centre O. The angle at O is 30°. Calculate the area of the sector. Give your answer in terms of π.
Medium
- A $(\frac{27}{4})\pi cm^{2}$
- B $(\frac{27}{2})\pi cm^{2}$
- C $27\pi cm^{2}$
- D $(\frac{9}{4})\pi cm^{2}$
7.The diagram shows a semicircle with diameter 9 cm. Calculate the total perimeter of this semicircle. $(Use \pi = 3.14)$
Medium
- A23.13 cm
- B14.13 cm
- C28.26 cm
- D18.13 cm
8.A circular pool has a diameter of 8 m. Calculate the circumference of the pool. $(Use \pi = 3.14)$
Medium
- A25.12 m
- B50.24 m
- C12.56 m
- D200.96 m

Summary

## 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).

Key terms

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; $\frac{\theta}{360} \times \pi r^2$.

Slides

Properties of a Circle

A circle is all points equidistant from a centre.
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.

What is the relationship between diameter and radius?

$d = \frac{r}{2}$
$d = r$
$d = 2r$
$d = \pi r$

Area and Circumference Formulae

Area of a circle: $A = \pi r^{2}$ .
Circumference: $C = \pi d$ or $C = 2\pi r$ .
These formulae are given in your exam.
Area is in square units, circumference in units of length.

The formula for the area of a circle is $A = \pi \times ____$ squared.

Arcs and Sectors

An arc is part of the circumference. Minor arc is smaller, major arc is larger.
A sector is a 'pizza slice' enclosed by two radii and an arc.
The angle at the centre is θ (θ).
The fraction of the circle is θ/360.

Which of these are true?

A sector is enclosed by two radii and an arc.
The major arc is the smaller arc.
The angle at the centre is always 90°.

Arc Length Formula

Arc length = (θ/360) $\times 2\pi r$ .
Step 1: Divide angle by 360 to get fraction.
Step 2: Find circumference of full circle (2πr).
Step 3: Multiply fraction by circumference.

What is the arc length of a sector with radius 5 cm and angle 72°? $(Use \pi = 3.14)$

2π cm
5π cm
10π cm
72π cm

Sector Area Formula

Area of sector = (θ/360) $\times \pi r^{2}$ .
Step 1: Divide angle by 360.
Step 2: Find area of full circle $(\pi r^{2})$ .
Step 3: Multiply fraction by area.

Area of sector $= (____/360) \times \pi r^{2}$ .

Working with Semicircles

A semicircle is half a circle (θ $= 180^{\circ})$ .
Area of semicircle $= (\frac{1}{2})\pi r^{2}$ .
Perimeter of semicircle $= (\frac{1}{2})\times 2\pi r +$ diameter $= \pi r + 2r$ .
Remember to include the straight edge (diameter) in perimeter.

What is the perimeter of a semicircle with radius 8 cm? (Leave in terms of π)

8π cm
16π cm
$(8\pi + 16)$ cm
$16\pi + 16$ cm

Finding Radius from Circumference

If circumference is known, radius $= C/(2\pi )$ .
Rearrange $C = 2\pi r$ to solve for r.
Example: $C = 56$ mm, $r = 56/(2\pi )$ ≈ 8.91 mm.

Given circumference C, radius $r = C / (2 \times ____)$

Finding Diameter from Circumference

Diameter $d = C/\pi$ .
Rearrange $C = \pi d$ to solve for d.
Example: $C = 4.25 m, d = 4.25/\pi$ ≈ $1.353 m = 135.3$ cm.

A circle has circumference 56 mm. What is its radius? $(Use \pi = 3.14)$

8.91 mm
17.83 mm
28 mm
112 mm

Exact Answers in Terms of π

Sometimes you are asked to give an answer 'in terms of π'.
Do not multiply by π; leave π in the answer.
Example: Area $= 32\pi cm^{2}$ , arc length $= 2\pi$ cm.

Which of these are written in terms of π?

$32\pi cm^{2}$
$100.48 cm^{2}$
2π cm

Flashcards

What is the formula for the area of a circle?

$A = \pi r^{2}$

What is the formula for the circumference of a circle?

$C = \pi d$ or $C = 2\pi r$

How do you find the radius from the diameter?

Radius = diameter $\div 2$

How do you find the diameter from the radius?

Diameter $= 2 \times$ radius

A circle has diameter 6 cm. Calculate its area.

Area $= \pi \times (3)^{2} = 9\pi cm^{2} (or 28.27 cm^{2})$

Calculate the circumference of a circle with radius 4.5 cm.

$C = 2\pi \times 4.5 = 9\pi$ cm (or 28.27 cm)

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

$C = \pi d = \pi \times 7 = 21.9911$ ... ≈ 21.99 cm

A circle has circumference 56 mm. Find its radius.

$r = C \div (2\pi ) = 56 \div (2\pi ) = 28/\pi$ ≈ 8.91 mm

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

$d = C \div \pi = 4.25 \div \pi$ ≈ $1.3528 m = 135.28$ cm

What is the formula for the area of a sector?

Area = (θ/360) $\times \pi r^{2}$

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

Arc length = (θ/360) $\times 2\pi r$

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

Area $= (\frac{30}{360}) \times \pi \times 9^{2} = (\frac{1}{12}) \times 81\pi = (\frac{27}{4})\pi cm^{2}$

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

Perimeter $= (\pi \times 9)/2 + 9 = 4.5\pi + 9$ cm (or 23.14 cm)

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

Circumference $= \pi \times 8 = 8\pi m$ (≈ 25.13 m); Area $= \pi \times 4^{2} = 16\pi m^{2}$ (≈ $50.27 m^{2})$

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

$C = 2\pi \times 11 = 22\pi$ 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 $= (\frac{140}{360}) \times 2\pi \times 11 = (\frac{7}{18}) \times 22\pi = (\frac{77}{9})\pi$ 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 $= 36\pi$ cm²; side of square $= 6\sqrt{2}$ cm, area $= 72$ cm²; shaded area $= 36\pi - 72 cm^{2}$ (≈ $41.10 cm^{2})$

Points P, Q, R on a semicircle of radius 8 cm, centre O, with angle $POQ = 90^{\circ}$ . Calculate the shaded area.

Shaded area = area of sector POQ - area of triangle $POQ = (\frac{90}{360})\times \pi \times 8^{2} - (\frac{1}{2})\times 8\times 8 = 16\pi - 32 cm^{2}$ (≈ $18.27 cm^{2})$

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 $= 2 \times 120 + \pi \times 60 = 240 + 60\pi m$ (≈ 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 $= 428.50 \div 1.4$ ≈ $306.07 s = 5$ 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 $= (\frac{1}{2})\pi \times 7^{2} - (\frac{1}{2})\pi \times 3^{2} = (\frac{1}{2})\pi (49-9) = 20\pi cm^{2}$ (≈ $62.83 cm^{2})$

The area of a circle is $76.4 cm^{2}$ . Find its radius.

$r = \sqrt{76.4/\pi }$ ≈ $\sqrt{24.32}$ ≈ 4.93 cm

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

$C = \pi \times 38$ $mm = 38\pi$ 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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