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Arcs & Sectors (OCR GCSE Maths)
Revision Note
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?
- 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
- The smaller sector is known as the minor 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 sector 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
Examiner Tip
- If you’re under pressure and can’t remember which formula is which, remember that area is always measured in square units (cm2, m2 etc.) so the formula with r2 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 cm2.
(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.
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.
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.
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