Arcs & Sectors (Edexcel IGCSE Further Pure Maths)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
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, , of an arc is
is the angle measured in degrees
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 is already 'built into' radians
The formula for the length, , of an arc is
is the angle measured in radians
is the radius
This is not on the exam formula sheet, so you need to remember it
Examiner Tips and Tricks
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
Use the arc length formula (in the degrees form) with and
(You could also convert to radians and use the radians version of the formula)
Round to 3 significant figures
(Note that cm is the exact value answer)
(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
The perimeter will include both the major arc and the two radii
Use the arc length formula (in the degrees form) with and
Round to 3 significant figures
(Note that 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 radians. Find the perimeter of the sector, giving your answer as an exact value.
Drawing a diagram can help
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 and
The question asks for an exact value answer, so just write that down (with units!)
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, , of a sector is
is the angle measured in degrees
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 is already 'built into' radians
The formula for the area, , of a sector is
is the angle measured in radians
is the radius
This is not on the formula sheet, so you need to remember it
Examiner Tips and Tricks
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 and
(You could also convert to radians and use the radians version of the formula)
Round to 3 significant figures
(Note that cm2 is the exact value answer)
(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 and
Round to 3 significant figures
(Note that cm2 is the exact value answer)
Worked Example
A sector of a circle has a radius of 7 cm and an angle at the centre of radians. Find the area of the sector, giving your answer as an exact value.
Drawing a diagram can help
Use the sector area formula (radians form) with and
The question asks for an exact value answer, so just write that down (with units!)
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