Arc Lengths & Sector Areas
What is an arc?
- An arc is a part of the circumference of a circle
- Two points on a circumference of a circle will create two arcs
- The smaller arc is known as the minor arc
- The bigger arc is known as the major arc
- The smaller arc is known as the minor arc
What is a sector?
- A sector is the part of a circle enclosed by two radii (radiuses) and an arc
- A sector looks like a slice of a circular pizza
- The curved edge of a sector is the arc
- Two radii in a circle will create two sectors
- The smaller sector is known as the minor sector
- The bigger sector is known as the major sector
- The smaller sector is known as the minor sector
What formulae do I need to know?
- You need to be able to calculate the length of an arc and the area of a sector
- The angle formed in a sector by the two radii is often labelled θ (the Greek letter “theta”)
- You can calculate the area of a sector or the length of an arc by adapting the formulae for the area or circumference of a circle
- A full circle is equal to 360° so the fraction will be the angle, θ°, out of 360°
- A full circle is equal to 360° so the fraction will be the angle, θ°, out of 360°
- Working with sector and arc formulae is just like working with any other formula:
- Write down what you know (or what you want to know)
- Pick the correct formula
- Substitute the values in and solve
How do I find the length of an arc?
- STEP 1
Divide the angle by 360 to form a fraction - STEP 2
Calculate the circumference of the full circle - STEP 3
Multiply the fraction by the circumference
How do I find the area of a sector?
- STEP 1
Divide the angle by 360 to form a fraction - STEP 2
Calculate the area of the full circle - STEP 3
Multiply the fraction by the area
Examiner Tip
- The area and circumference of a circle formulae are given to you in the exam.
- You just need to remember how to find the correct fraction of the whole circle.
Worked example
A sector of a circle is shown.
The angle, θ, is 72° and the radius, r, is 5 cm.
(a)
Find the area of the sector, giving your answer correct to 3 significant figures.
Substitute θ = 72° and r = 5 into the formula for the area of a sector,
Use a calculator to work out this value
15.70796...
Round your answer to 3 significant figures
15.7 cm2
(b)
Find the length of the arc of the sector, giving your answer as a multiple of .
Substitute θ = 72° and r = 5 into the formula for the length of an arc,
Simplify the number part without
Write down the final answer with
cm