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
What is a sector?
In technical terms, a sector is the part of a circle enclosed by two radii (radiuses) and an arc
It’s much easier to think of a sector as the shape of a slice of a circular pizza (or cake, or pie, or …) and an arc as the curvy bit at the end of it (where the crust is)
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
If the angle of the slice is θ (the Greek letter “theta”) then the formulae for the area of a sector and the length of an arc are just fractions of the area and circumference of a circle:
Remember that a full circle is equal to 360° so the fraction will be the angle, out of 360
If you are not too good at remembering formulae there is a logic to these two
You’ll need to remember the circumference and area formulas
After that we are just finding a fraction of the whole circle – “θ out of 360”
Working with s ector and arc formulae is just like working with any other formula
WRITE DOWN – what you know (what you want to know)
Pick correct FORMULA
SUBSTITUTE and SOLVE
Exam Tip
If you’re under pressure and can’t remember which formula is which, remember that area is always measured in square units (cm^{2}, m^{2} etc.) so the formula with r^{2} in it is the one for area
The length of an arc is just a length, so its units will be the same as for length (cm, m, etc)
Worked example
AOB is a sector of a circle with angle 42°, as shown.
The area of the sector AOB is 28 cm ^{2}.
(a)
Find the radius of the circle, giving your answer correct to 2 decimal places.
We know the area and the angle and want to find the radius so we will need to substitute the information into the formula for the area of a sector and solve to find the radius.
Substitute A = 28 and θ = 42° into the formula for the area of a sector,
.
Simplify.
Rearrange to find
.
Round to 2 decimal places. The second decimal place has the value 4 and the third decimal place is a 0, which is less than 5 so do not change the value of the second decimal place.
(b)
Find the length of the arc AB, giving your answer correct to the nearest whole number.
We know the radius and the angle so we can substitute the information into the formula for the length of an arc.
Substitute r = 8.7403… and θ = 42° into the formula for the length of an arc,
.
Use your calculator to find the answer, type the radius in carefully or use the memory function on your calculator.
Round to the nearest whole number. The units place has the value 6 and the first decimal place is a 4, which is less than 5 so do not change the value of the units.