Circles, Arcs and Sectors
Circle measure combines precise vocabulary with proportional reasoning. This lesson develops circumference and area, then uses fractions of a full turn for arcs, sectors, composite boundaries and shaded regions.
Foundation & Higher. Arc length, sector area and positive reverse calculations belong in the shared route. Circle theorems, circle equations and radians are different topics and are not taught here.
Circle language
The centre is the fixed point. A radius joins the centre to the circumference. A diameter passes through the centre from one side to the other, so d=2r. A chord joins two points on the circumference. A tangent touches the circle at one point.
An arc is part of the circumference. A sector is the region between two radii and their arc. A segment is the region between a chord and its arc.
Circumference
The circumference is the distance around a circle: C=2πr, or equivalently C=πd.
Worked example: radius 6 cm gives C=2π×6=12π cm, approximately 37.7 cm to 1 decimal place.
If an exact answer is requested, leave π in the answer. If a decimal is required, keep π on the calculator until the final rounding step.
Area of a circle
The area enclosed by a circle is A=πr². The radius is squared; the diameter is not substituted directly unless you first halve it.
Worked example: diameter 10 m means radius 5 m. A=π×5²=25π m²≈78.5 m².
Notice the units: circumference is a length, while area uses square units.
Arc length — a fraction of the circumference
A full turn is 360°. A sector with central angle θ therefore contains θ/360 of the circle. Its arc length is (θ/360)×2πr.
Worked example: r=9 cm and θ=80°. Arc=(80/360)×18π=4π cm≈12.6 cm.
This is proportional reasoning: halving the central angle halves the arc length when the radius stays fixed.
Sector area
The same fraction-of-a-turn idea applies to area: sector area=(θ/360)×πr².
Worked example: a 120° sector of radius 6 cm has area (120/360)×36π=12π cm²≈37.7 cm².
For a major sector, use its central angle directly, or subtract the minor sector from the full circle.
Sector perimeter includes the two radii
Arc length is only the curved part. The perimeter of an ordinary sector is arc length + 2r.
Worked example: a 90° sector has radius 8 cm. Arc=(90/360)×16π=4π cm. Perimeter=16+4π cm≈28.6 cm.
Working backwards
Circle formulae can be rearranged. If C=30π cm, then 2πr=30π, so r=15 cm.
If a sector of radius 10 cm has area 25π cm², then (θ/360)×100π=25π. Therefore θ/360=1/4 and θ=90°.
Keep exact π terms as long as possible because they often cancel cleanly in reverse calculations.
Composite circles, annuli and shaded regions
Semicircles and quadrants are simply 1/2 and 1/4 of a circle. For an annulus — a ring between concentric circles — subtract the inner area from the outer area.
Worked example: outer radius 7 cm, inner radius 4 cm. Ring area=49π−16π=33π cm².
For composite perimeters, trace only the boundary that is actually exposed. Include straight edges as well as arcs where appropriate.
Sector or segment?
A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc. Do not use the sector-area formula for a segment without accounting for the triangle between the radii.
When that triangle area is already accessible, segment area = sector area − triangle area. General trigonometric segment calculations are reserved for the later non-right-angled trigonometry lesson.
Think Like a Mathematician
Arc and sector formulae are not really new formulae: both are the same fraction θ/360 of a whole-circle quantity. Start from the full circle, identify the fraction of a turn, and the method becomes easier to reconstruct.
Common Mistakes
Diameter for radius: halve the diameter before using A=πr².
Arc versus sector perimeter: sector perimeter also includes the two straight radii.
Wrong fraction: degrees are a fraction of 360°, not 180°.
Rounding too early: keep π and unrounded calculator values until the final step.
Practice — circles, arcs and sectors
1. A circle has radius 5 cm. Find its circumference exactly.
10π cm.
2. A circle has diameter 14 m. Find its area exactly.
r=7, so 49π m².
3. Find the arc length of a 60° sector of radius 12 cm.
(60/360)×24π=4π cm.
4. Find the area of a 90° sector of radius 8 cm.
(90/360)×64π=16π cm².
5. Find the perimeter of that 90° sector.
Arc=4π cm, so perimeter=16+4π cm.
6. A circle has circumference 18π cm. Find its radius.
2πr=18π, so r=9 cm.
7. A sector of radius 6 cm has area 9π cm². Find its central angle.
(θ/360)×36π=9π, so θ=90°.
8. An annulus has outer radius 10 cm and inner radius 6 cm. Find its area exactly.
100π−36π=64π cm².
9. Explain the difference between a sector and a segment.
A sector is bounded by two radii and an arc; a segment by a chord and an arc.
10. A semicircle has radius 4 cm. Find the perimeter of the whole semicircular shape.
Half circumference=4π cm; add diameter 8 cm, giving 8+4π cm.