MISSION · Geometry
Arc Length and Sector Area
Every circle formula, scaled down by the fraction of a turn you kept
LEVEL 1 · EXPLORE
Make the idea move
Change the controls and watch what happens. Follow your curiosity and see what you notice.
Loading the activity…
LEVEL 2 · LEARN
Unlock the explanation
Open the chapters in any order. Try the examples as you go.
The idea in one sentence
A sector is a fraction of a circle, so its arc and its area are the circle’s circumference and area multiplied by that same fraction.
The fraction is all there is
A full turn is 360°. An angle of \(\theta\) degrees at the centre keeps \(\dfrac{\theta}{360}\) of the circle, so:
\[\text{arc length} = \frac{\theta}{360} \times 2\pi r \qquad \text{sector area} = \frac{\theta}{360} \times \pi r^2\]A 90° sector of a radius-8 circle is a quarter circle. Quarter of \(64\pi\) is \(16\pi\). You could do that without either formula, and checking a formula against a case you can reason out is the fastest way to know you’ve written it correctly.
There is nothing else to memorise here. Both formulas are “the whole thing, times the fraction you kept”.
Radians make it cleaner
A radian is the angle that makes the arc exactly as long as the radius. Since the whole circumference is \(2\pi r\), a full turn is \(2\pi\) radians, so \(\pi\) radians is 180°.
In radians the fraction is \(\dfrac{\theta}{2\pi}\), and the formulas collapse:
\[s = r\theta \qquad A = \tfrac{1}{2}r^2\theta\]No 360, no \(\pi\) in the working. This is the actual reason radians exist — not to make trigonometry harder, but because \(s = r\theta\) is the simplest possible statement of what an angle measures. Calculus later insists on radians for the same underlying reason.
Convert by multiplying: degrees to radians, × \(\pi/180\); radians to degrees, × \(180/\pi\).
The segment
A segment is the region between a chord and the arc — a sector with the triangle cut off. So:
\[\text{segment} = \tfrac{1}{2}r^2\theta - \tfrac{1}{2}r^2\sin\theta = \tfrac{1}{2}r^2(\theta - \sin\theta)\]The triangle formed by the two radii and the chord has area \(\frac{1}{2}ab\sin C\), which here is \(\frac{1}{2}r^2\sin\theta\). Subtract it from the sector and the segment is what’s left.
Perimeter of a sector
The perimeter of a pizza slice is not just the arc. It’s the arc plus two radii:
\[P = r\theta + 2r\]Questions ask for this specifically because it’s easy to answer the wrong one.
Where it shows up
Angular speed, the distance a wheel travels in a part-turn, radar sweeps, the area a sprinkler covers, the length of a curved road. Any time something goes round a fixed centre through part of a turn, these two formulas are what you need.
Common mix-ups
- Mixing units. Radian formulas with degree angles give nonsense. Check your calculator’s mode.
- Confusing arc length with sector area. One is a length, one is an area — a stray \(r\) is the difference, and the units tell you which you’ve got.
- Giving the arc when the question asks for the perimeter, forgetting the two radii.
- Rounding π too early, which shows up badly when the answer is meant to be exact.
Check yourself
Find the area of a 90° sector with radius 8, then the arc length for an angle of 2 radians at radius 5.
A quarter of π × 64 is 16π. In radians, s = rθ = 5 × 2 = 10 — no π needed, which is the point of radians.
LEVEL 3 · REFLECT
Add this quest to your trail
When you feel ready, mark this idea as explored. You can always come back and try it again.
EXPLORER STRATEGIES