Arcs & Sectors (DP IB Maths: AA HL)

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Amber

Last updated

12 February 2024

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Length of an Arc

What is an arc?

An arc is a part of the circumference of a circle
- It is easiest to think of it as the crust of a single slice of pizza
The length of an arc depends on the size of the angle at the centre of the circle
If the angle at the centre is less than 180° then the arc is known as a minor arc
- This could be considered as the crust of a single slice of pizza
If the angle at the centre is more than 180° then the arc is known as a major arc
- This could be considered as the crust of the remaining pizza after a slice has been taken away

How do I find the length of an arc?

The length of an arc is simply a fraction of the 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

$l = \frac{θ}{360} \times 2 π r$

- Where $θ$ is the angle measured in degrees
- $r$ is the radius
- This is in the formula booklet for radian measure only
  - Remember 2π radians = 360°

Examiner Tip

Make sure that you read the question carefully to determine if you need to calculate the arc length of a sector, the perimeter or something else that incorporates the arc length!

Worked example

A circular pizza has had a slice cut from it, the angle of the slice that was cut was 38 °. The radius of the pizza is 12 cm. Find

the length of the outside crust of the slice of pizza (the minor arc), ai-sl-3-1-2-length-arc-we-i

ii)

the perimeter of the remaining pizza. ai-sl-3-1-2-length-arcwe-ii

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Area of a Sector

What is a sector?

A sector is a part of a circle enclosed by two radii (radiuses) and an arc
- It is easier to 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
If the angle at the centre is less than 180° then the sector is known as a minor sector
- This could be considered as the shape of a single slice of pizza
If the angle at the centre is more than 180° then the sector is known as a major sector
- This could be considered as the shape of the remaining pizza after a slice has been taken away

How do I find the area of a sector?

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

$A = \frac{θ}{360} \times π r^{2}$

- Where $θ$ is the angle measured in degrees
- $r$ is the radius
- This is in the formula booklet for radian measure only
  - Remember 2π radians = 360°

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. Find

the area Jamie will paint blue,

ai-sl-3-1-2-area-sectorwe-i

ii)

the area Jamie will paint yellow. ai-sl-3-1-2-area-sectorweii

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Arcs & Sectors Using Radians

How do I use radians to find the length of an arc?

As the radian measure for a full turn is $2 π$ , the fraction of the circle becomes $\frac{θ}{2 π}$
Working in radians, the formula for the length of an arc will become

$l = \frac{θ}{2 π} \times 2 π r$

Simplifying, 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 given in the formula booklet, you do not need to remember it

How do I use radians to find the area of a sector?

As the radian measure for a full turn is $2 π$ , the fraction of the circle becomes $\frac{θ}{2 π}$
Working in radians, the formula for the area of a sector will become

$A = \frac{θ}{2 π} \times π r^{2}$

Simplifying, the formula for the area, $A$ , of a sector is

$A = \frac{1}{2} r^{2} θ$

- $θ$ is the angle measured in radians
- $r$ is the radius
- This is given in the formula booklet, you do not need to remember it

Worked example

A slice of cake forms a sector of a circle with an angle of $\frac{π}{6}$ radians and radius of 7 cm. Find the area of the surface of the slice of cake and its perimeter.

aa-sl-3-1-3-arcs-and-sectors-using-radians-we-solution