Circles, arcs and sectors
Section: Mensuration | Syllabus: Cambridge IGCSE Mathematics (0580)
Circumference and Area of a Circle
The circumference and area of a circle are calculated using its radius or diameter, and the constant π (pi).
- Circumference =π d=2π r
- Area =π r^2
- π is a constant, approximately 3.14159 - use the π button on a calculator for the most accurate answers, unless told to use a given approximation
Worked Example: Finding the Diameter from a Circumference
- Question: A circular track has a circumference of 75.4 m. Find its diameter, correct to 3 significant figures.
- Step 1: C=π d, so d=(C/π)
- Step 2: d=(75.4/π)
- Answer: d ≈ 24.0 m (3 s.f.)
Worked Example: Finding the Area of a Circle
- Question: A circle has a radius of 9 cm. Find its area, in terms of π.
- Step 1: π9^2
- Step 2: =81π
- Answer: 81π cm²
Common Mistakes
MistakeUsing the diameter in the area formula instead of the radius, e.g. writing area = πd²
Fixthe area formula always uses the radius (area = πr²) - if only the diameter is given, halve it first
MistakeRounding π too early in a multi-step calculation, causing a less accurate final answer
Fixkeep the exact π symbol (or the full calculator value) throughout the working, and round only the final answer
Arc Length
An arc is a part of a circle's circumference. Its length is a fraction of the full circumference, matching the fraction that the arc's angle is of a full turn.
- Arc length =(angle at centre/360)×π d
- The angle used is the angle at the centre between the two radii bounding the arc
- Always check whether the question wants the minor arc (the shorter one) or the major arc (the longer one)
Worked Example: Finding an Arc Length
- Question: A sector has a radius of 10 cm and an angle at the centre of 72°. Find the length of the arc, in terms of π.
- Step 1: (72/360)×π20
- Step 2: =0.220π
- Step 3: =4π
- Answer: 4π cm
Common Mistakes
MistakeUsing just the radius inside the arc length formula, forgetting the fraction is applied to the whole circumference
Fixmultiply the angle fraction by the full circumference (πd or 2πr), not just the radius alone
MistakeForgetting to correctly identify which fraction of 360° the given angle represents
Fixwrite the fraction as angle/360° first, then simplify if possible, before multiplying
Sector Area
A sector is the region enclosed by two radii and an arc. Its area is a fraction of the full circle's area, matching the fraction that the sector's angle is of a full turn.
- Sector area =(angle at centre/360)×π r^2
- This uses the same fraction idea as arc length, but applied to the full circle's area instead of its circumference
- Always identify the angle at the centre for the specific sector required before substituting into the formula
Worked Example: Finding a Sector Area
- Question: A sector has a radius of 8 cm and an angle at the centre of 45°. Find its area, in terms of π.
- Step 1: (45/360)×π8^2
- Step 2: =(1/8)64π
- Step 3: =8π
- Answer: 8π cm²
Common Mistakes
MistakeUsing the diameter instead of the radius when substituting into πr²
Fixthe sector area formula uses r², the radius squared - convert from diameter to radius first if needed
MistakeForgetting to square the radius, treating the formula as (angle/360)×π×r instead of ×r²
Fixalways square the radius before multiplying by π and the angle fraction
Major and Minor Sectors
Extended Only
Every pair of radii divides a circle into two sectors - a smaller minor sector and a larger major sector. Their angles always add up to 360°.
- The minor sector has the smaller angle at the centre; the major sector has the larger (often reflex) angle
- The two sector angles always sum to 360°, so the major sector's angle = 360° − minor sector's angle
- Once the correct angle is found, the same sector area (or arc length) formula applies as for any other sector
Worked Example: Finding the Area of a Major Sector
- Question: A circle has centre O and radius 6 cm. The minor sector has an angle of 120° at the centre. Find the area of the major sector, in terms of π, in its simplest form.
- Step 1: The major sector's angle: 360-120=240°
- Step 2: (240/360)×π6^2
- Step 3: Simplify the fraction: (240/360)=(2/3)
- Step 4: (2/3)36π=24π
- Answer: 24π cm²
Common Mistakes
MistakeUsing the minor sector's angle when the major sector's area is actually required, or vice versa
Fixalways check carefully which sector (major or minor) the question asks for, and subtract from 360° if needed before calculating
MistakeNot simplifying the angle fraction before multiplying, leading to messier arithmetic
Fixsimplify a fraction like 240/360 to 2/3 first - this often makes the multiplication by πr² much easier to manage
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