Arcs & Sectors (Edexcel IGCSE Further Maths)

Revision Note

Author

Roger

Expertise

Maths

Length of an Arc

What is an arc?

An arc is a part of the circumference of a circle
- You can think of it as the crust on a slice of pizza
The length of an arc depends on
- the size of the angle at the centre of the circle
- the radius of the circle
If the angle at the centre is less than 180° then the arc is known as a minor arc
If the angle at the centre is more than 180° then the arc is known as a major arc

How do I find the length of an arc using degrees?

The length of an arc is simply a fraction of the total circumference of a circle
- The fraction can be found by dividing the angle at the centre by 360°
The formula for the length, $l$ , of an arc is
- $l = \frac{θ}{360} \times 2 π r$

- - $θ$ is the angle measured in degrees
  - $r$ is the radius
- This is not on the exam formula sheet, so you need to remember it

How do I find the length of an arc using radians?

When working in radians the formula for arc length is much simpler
- This is because $π$ is already 'built into' radians
The formula for the length, $l$ , of an arc is
- $l = r θ$

- - $θ$ is the angle measured in radians
  - $r$ is the radius
- This is not on the exam formula sheet, so you need to remember it

Exam Tip

Be careful on arc length questions
- Finding the arc length may only be part of a larger question
- For example finding the total perimeter of a circle sector
Make sure you are using the formula that matches the angle measure (degrees or radians)

Worked example

A circular pizza has had a slice cut from it. The slice is in the shape of a sector, with the angle at the centre being 38°. The radius of the pizza is 12 cm.

(a)

Find the length of the outside crust of the slice of pizza (the minor arc), giving your answer correct to 3 significant figures.

Drawing a diagram can help with questions like this

Minor sector with minor arc, angle and radius

Use the arc length formula (in the degrees form) with

r = 12

and

θ = 38

(You could also convert

38 °

to radians and use the radians version of the formula)

l = \frac{38}{360} \times 2 π (12) = \frac{38 π}{15} = 7.958701 . . .

Round to 3 significant figures

(Note that

\frac{38 π}{15}

cm is the exact value answer)

7.96 cm (3 s . f .)

(b)

Find the perimeter of the remaining pizza, giving your answer correct to 3 significant figures.

Drawing a diagram can help

The remaining pizza will be in the shape of a major sector

The angle at the centre will be

360 - 38 = 322 °

Major sector of circle with radius and angle at centre

The perimeter will include both the major arc and the two radii

Use the arc length formula (in the degrees form) with

r = 12

and

θ = 322

perimeter = (\frac{322}{360} \times 2 π (12)) + 12 + 12 = \frac{322 π}{15} + 24 = 91.439522 . . .

Round to 3 significant figures

(Note that

\frac{322 π}{15} + 24

cm is the exact value answer)

$91.4 cm (3 s . f .)$

Worked example

A sector of a circle has a radius of 7cm and an angle at the centre of $\frac{π}{6}$ radians. Find the perimeter of the sector, giving your answer as an exact value.

Drawing a diagram can help

Circle sector with radius and angle at centre

The perimeter of sector will include both the length of the arc and the two radii

Use the arc length formula (in the radians form) with $r = 7$ and $θ = \frac{π}{6}$

$perimeter = (7) (\frac{π}{6}) + 7 + 7 = \frac{7 π}{6} + 14$

The question asks for an exact value answer, so just write that down (with units!)

$\frac{7 π}{6} + 14 cm$

Area of a Sector

What is a sector?

A sector is a part of a circle enclosed by two radii (radiuses) and an arc
- You can think of this as the shape of a single slice of pizza
The area of a sector depends on
- the size of the angle at the centre of the sector
- the radius of the circle
If the angle at the centre is less than 180° then the sector is known as a minor sector
If the angle at the centre is more than 180° then the sector is known as a major sector

How do I find the area of a sector using degrees?

The area of a sector is simply a fraction of the area of the whole circle
- The fraction can be found by dividing the angle at the centre by 360°
The formula for the area, $A$ , of a sector is
- $A = \frac{θ}{360} \times π r^{2}$

- - $θ$ is the angle measured in degrees
  - $r$ is the radius
- This is not on the formula sheet, so you need to remember it

How do I find the area of a sector using radians?

When working in radians the formula for sector area is much simpler
- This is because $π$ is already 'built into' radians
The formula for the area, , of a sector is
- $A = \frac{1}{2} r^{2} θ$

- - $θ$ is the angle measured in radians
  - $r$ is the radius
- This is not on the formula sheet, so you need to remember it

Exam Tip

Make sure you are using the formula that matches the angle measure (degrees or radians)

Worked example

Jamie has divided a circle of radius 50 cm into two sectors: a minor sector of angle 100°, and a major sector of angle 260°. He is going to paint the minor sector blue and the major sector yellow.

(a)

Find the area Jamie will paint blue, giving your answer correct to 3 significant figures

Drawing a diagram can be helpful in questions like this

Use the sector area formula (degrees form) with

r = 50

and

θ = 100

(You could also convert

100 °

to radians and use the radians version of the formula)

A = \frac{100}{360} \times π {(50)}^{2} = \frac{6250 π}{9} = 2181.6615 . . .

Round to 3 significant figures

(Note that

\frac{6250 π}{9}

cm² is the exact value answer)

2180 {cm}^{2} (3 s . f .)

(b)

Find the area Jamie will paint yellow, giving your answer correct to 3 significant figures

This will be the rest of the circle from the preceding diagram

Use the sector area formula (degrees form) with

r = 50

and

θ = 260

A = \frac{260}{160} \times π {(50)}^{2} = \frac{16250 π}{9} = 5672.3200 . . .

Round to 3 significant figures

(Note that

\frac{16250 π}{9}

cm² is the exact value answer)

5670 {cm}^{2} (3 s . f .)

Worked example

A sector of a circle has a radius of 7 cm and an angle at the centre of $\frac{π}{6}$ radians. Find the area of the sector, giving your answer as an exact value.

Drawing a diagram can help

Circle sector with radius and angle a centre

Use the sector area formula (radians form) with $r = 7$ and $θ = \frac{π}{6}$

$A = \frac{1}{2} {(7)}^{2} (\frac{π}{6}) = \frac{49 π}{12}$

The question asks for an exact value answer, so just write that down (with units!)

$\frac{49 π}{12} {cm}^{2}$

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Arcs & Sectors (Edexcel IGCSE Further Maths)

Revision Note

What is an arc?

How do I find the length of an arc using degrees?

How do I find the length of an arc using radians?

What is a sector?

How do I find the area of a sector using degrees?

How do I find the area of a sector using radians?

You've read 0 of your 0 free revision notes

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Author: Roger