A sector is a slice of pizza. A segment is what is left after you bite the crust off straight.
Find the area
A sector has a radius of 12 cm and a central angle of 60°. What is its area?
60° is a sixth of 360°, and the whole circle has area π(12)² = 144π cm². A sixth of that is 24π cm². D is the whole circle, forgetting to take the fraction. B is the arc length, 4π cm — a length, not an area, and the units give it away.
1
Everything is a fraction of the whole circle.
The fraction is always θ360.
Wanted
Whole circle
Fraction of it
Arc length
2πr
(θ360) × 2πr
Sector area
πr²
(θ360) × πr²
A sector is the slice bounded by two radii and the arc between them. A segment is the region between a chord and its arc. With π taken as 3.14, a whole circle of radius 3 cm has area 3.14 × 3² = 3.14 × 9 = 28.26 cm².
There is nothing new to memorise. Work out the whole circle, then take the slice. Common angles help: 90° is a quarter, 60° a sixth, 45° an eighth, 120° a third.
2
Arc length is a distance; sector area is a surface.
Same fraction, different starting quantity.
r = 12, θ = 60°
the opening sector
fraction = 60360 = 16
simplify first, always
arc = (16)(2π · 12) = 4π cm
a length
area = (16)(π · 144) = 24π cm²
an area
Let the units check your work
An arc length is in cm, a sector area in cm².
If a question asks for area and your answer used 2πr, the units will not match —
and that is the fastest self-check in the whole topic.
3
A segment is a sector with the triangle removed.
segment = sector − triangle
r = 14, θ = 90°
a quarter circle
sector = (14)π(196) = 49π
≈ 153.9
triangle = (12)(14)(14) = 98
the two radii are perpendicular
segment = 49π − 98 ≈ 55.9
cm²
49π − 98 ≈ 55.9 cm²
The answer must be smaller than the sector and bigger than zero. Getting a negative
means the triangle and sector were subtracted the wrong way round.
4
When the angle is not a friendly one.
The triangle needs the sine formula once the angle stops being 90°.
triangle = (12)r² sin θ
two sides and the angle between
r = 8, θ = 120°
sin 120° ≈ 0.866
sector = (13)π(64) ≈ 67.0
120360 = 13
triangle = (12)(64)(0.866) ≈ 27.7
not a right triangle
segment ≈ 39.3 cm²
67.0 − 27.7
Note that sin 120° is positive and equal to sin 60°. An obtuse central angle still gives a perfectly ordinary positive triangle area.
5
Working backwards to r or θ.
Givensector area 12π, radius 6find θ
Set up(θ360)π(36) = 12ππ cancels
Solve36θ = 4320θ = 120° ✓
Sensible? 12π out of a whole circle of 36π is a third, and a third of 360° is 120° ✓
Spotting the fraction directly is often faster than the algebra.
Remember the flow
Find θ360 · Multiply by the whole circle · Segment? subtract the triangle · Check the units
Length uses 2πr, area uses πr². Mixing them is the error the units always catch.
6
Variations: π as 227, the radius unknown, the perimeter of a sector.
The same θ360 fraction; each case asks for something slightly different.
π as 227: a 45° arc in a circle of radius 28 cm
2 × 227 × 28 = 176 cm
the full circumference; 28 ÷ 7 = 4
arc = 176 ÷ 8 = 22 cm
45° is an eighth
Different: 227 is chosen when r is a multiple of 7, so the sevens cancel and the answer is exact.
The radius is unknown: a 45° arc measures 5π cm
(18)(2πr) = 5π
set up the arc formula
2πr = 40π, r = 20 cm
check: (18)(40π) = 5π ✓
Different: θ is known and r is the unknown; π cancels just as it did for θ.
The perimeter of a sector: r = 9 cm, θ = 80°
arc = (80360)(18π) = 4π
80360 = 29
perimeter = 4π + 9 + 9
the arc plus BOTH radii
= 4π + 18 ≈ 30.57 cm
a length, so cm
Different: the perimeter is not just the arc — the two straight edges are part of the boundary too.
On the exam
Expect an arc length, a sector area, a segment area, or a backwards item that gives the area and asks for θ or r, often with π taken as 3.14 or 227. The fastest route is to write θ360 as a simple fraction first, then multiply the whole circle. The 10-second check: the units (cm for a length, cm² for an area), and a sector must be smaller than the whole circle by exactly that fraction.
Now you try
Practice problem
Find the segment area
A circle has radius 6 cm. A chord cuts off a sector with a central angle of 90°. What is the area of the segment between the chord and the arc?
The sector is a quarter circle: (14)π(36) = 9π cm². The triangle has two perpendicular radii, so its area is (12)(6)(6) = 18 cm². The segment is what is left: 9π − 18 ≈ 10.3 cm². A is the sector, with the triangle still in it, and C adds the triangle instead of removing it.
Hint: at 90° the two radii are perpendicular, so the triangle is
right-angled and its area is simply half the product of the radii.
