Arcs and Sectors of a Circle
“A sector is just 'whole circle × my slice's fraction (θ/360)'.”
The formula
π r² × (θ / 360)How to read it: a sector's area is the whole circle's area times the fraction the central angle takes up (θ/360)
- r
- — the radius of the circle
- θ
- — the sector's central angle (in degrees)
- θ / 360
- — the fraction of the whole circle the sector takes up
The hook
A sector is really just 'a slice of pizza' — all you need is what fraction of the whole your slice is.
In plain words
A sector is part of a circle. Its central angle θ takes up some fraction of a full turn (360°), and it grabs that same fraction of the circle's total area (or circumference).
The intuition
Picture a whole pizza. A 90° central angle is a quarter of a full turn, so both the area and the arc length are exactly a quarter of the whole. θ/360 is precisely 'the fraction that is my slice'.
How it's built
Two parts — ① the circle's total amount (area πr², circumference 2πr), ② your slice's fraction θ/360. Multiply them: sector area = πr²·(θ/360), arc length = 2πr·(θ/360).
Example
A sector with radius 6 and central angle 90°: area = π·6²·(90/360) = 36π·(1/4) = 9π. Arc length = 2π·6·(1/4) = 3π.
Common mistake
It's easy to plug the diameter in for the radius, or to forget the θ/360 factor and answer with the whole circle's area. Nail down 'what fraction' first.
Where it's used
Pizza and pie slices, the area a clock hand sweeps, designing circular tracks, fans, and blades — anywhere a portion of a circle appears.
Where it came from
Archimedes found the circle's area πr² by slicing a circle into infinitely many pieces; the sector is the natural extension of dividing that circle by an angle's fraction.
Quick check
What is the area of a sector with radius 6 and central angle 90°? (answer as a multiple of π)
- 3π
- 6π
- 9π✓
- 36π
Practice
What is the arc length of a sector with radius 6 and central angle 90°? Answer as the coefficient of π (e.g. if it's 3π, answer 3).
Answer: 3
- arc length = 2πr × (θ/360) = 2π·6 × (90/360)
- = 12π × (1/4) = 3π
- the coefficient of π is 3
Takeaway: Arc length is the 'circumference 2πr' times the slice's fraction.
What is the area of a whole circle with radius 5? Answer as the coefficient of π.
Answer: 25
- area of a circle = πr² = π·5²
- = 25π
- the coefficient of π is 25
Takeaway: At a central angle of 360° (a full turn), the sector is the whole circle.
Find the area of a sector with radius 4 and central angle 90°, as the coefficient of π. Also write on your notepad what fraction of the whole circle it is.
Answer: 4
- 90° is 1/4 of 360°
- area = π·4² × (1/4) = 16π × (1/4) = 4π
- the coefficient of π is 4
Takeaway: Pin down 'what fraction' with θ/360 first and the computation gets easy.