Arc Length & Area of Sectors
When a pizza chef cuts a slice, they create a sector — a wedge-shaped piece of a circle. But how much crust is on the curved edge? And what is the area of the topping? In this lesson you will learn the elegant formulas that answer both questions, and discover why radians make them beautifully simple.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
A pizza has radius 20 cm. A slice is cut with an angle of $45^\circ$ at the centre. Without using any formulas — estimate the arc length of the crust on this slice. Then consider how you might calculate it exactly if you knew a full circle's circumference is $2\pi r$.
There are only two formulas in this entire lesson — and they both emerge from one idea. When $\theta$ is measured in radians, the formulas for arc length and sector area collapse into their simplest possible forms.
Every sector problem in this module travels along one of two roads: arc length uses $l = r\theta$, and sector area uses $A = \frac{1}{2}r^2\theta$. Both require $\theta$ in radians. Convert first, calculate second.
$A = \tfrac{1}{2}r^2\theta$
Key facts
- The formulas for arc length and sector area in radians
- How to convert degree angles to radians before using the formulas
- The relationship between arc length, radius, and angle
Concepts
- Why the radian formulas are simpler than the degree formulas
- How sector area relates to the proportion of a full circle
- The connection between arc length and the definition of a radian
Skills
- Calculate arc length given radius and central angle
- Calculate sector area given radius and central angle
- Solve for unknown radius or angle given arc length or area
- Apply arc length and sector area to real-world problems
When an angle $\theta$ is measured in radians, the arc length $l$ swept out by that angle in a circle of radius $r$ is given by:
This formula is elegant because it follows directly from the definition of a radian: $\theta = \frac{l}{r}$. In degrees, the equivalent formula is $l = 2\pi r \times \frac{\theta}{360^\circ}$ — messier, and easy to misapply.
A sector is the "pizza slice" region bounded by two radii and an arc. Its area comes from taking the same fraction of the full circle's area ($\pi r^2$) as the angle is of a full revolution ($2\pi$ radians). When you simplify $\pi r^2 \times \frac{\theta}{2\pi}$, the $\pi$ cancels and you get:
Circle sector with radius r, central angle θ, and arc length l — radian formulas are cleaner because radians are the natural unit.
If you know the arc length or sector area, you can rearrange to find the missing quantity:
- Find angle from arc length: $\theta = \frac{l}{r}$
- Find radius from arc length: $r = \frac{l}{\theta}$
- Find angle from sector area: $\theta = \frac{2A}{r^2}$
- Find radius from sector area: $r = \sqrt{\frac{2A}{\theta}}$
Arc length: $l = r\theta$ where $\theta$ is in radians — rearranges to $\theta = \frac{l}{r}$ or $r = \frac{l}{\theta}$; Sector area: $A = \frac{1}{2}r^2\theta$ — comes from $\pi r^2 \times \frac{\theta}{2\pi}$, the $\pi$ cancels
Pause — copy the arc length formula $l = r\theta$ and the sector area formula $A = \frac{1}{2}r^2\theta$ (with the $\pi$-cancellation derivation) into your book.
Worked examples · 3 in a row, reveal as you go
Find the arc length of a sector with radius 8 cm and central angle $\frac{\pi}{4}$ radians.
A sector has radius 10 cm and central angle $60^\circ$. Find its area.
A sector has area $27\pi$ cm$^2$ and central angle $\frac{\pi}{2}$ radians. Find the radius.
Common errors · the 3 traps that cost marks
Quick-fire practice · 5 reps
$r = 6$ cm, $\theta = \frac{\pi}{3}$ rad. Find $l$.
$r = 10$ cm, $\theta = \frac{3\pi}{4}$ rad. Find $A$.
$r = 12$ cm, $\theta = 45^\circ$. Find $l$ and $A$.
$l = 8\pi$ cm, $r = 4$ cm. Find $\theta$ in radians.
$A = 50\pi$ cm$^2$, $\theta = \frac{\pi}{2}$ rad. Find $r$.
Earlier you were asked: A pizza has radius 20 cm and a slice angle of $45^\circ$. Estimate the arc length of the crust.
A full circle has circumference $2\pi r = 40\pi \approx 125.7$ cm. The slice is $\frac{45}{360} = \frac{1}{8}$ of the full pizza. So the arc length is $\frac{1}{8}$ of the circumference: $\frac{40\pi}{8} = 5\pi \approx 15.7$ cm. In radians, $45^\circ = \frac{\pi}{4}$, so $l = r\theta = 20 \times \frac{\pi}{4} = 5\pi$ cm. The radian formula gives the same answer instantly without needing to calculate the full circumference first.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. A sector has radius 9 cm and central angle $\frac{2\pi}{3}$ radians. (a) Find the exact arc length. (b) Find the exact area of the sector. 3 MARKS
Q2. A piece of wire 40 cm long is bent to form the perimeter of a sector. If the radius of the sector is 12 cm, find the angle of the sector in radians. 3 MARKS
Q3. Two sectors have the same area. Sector A has radius 6 cm and angle $\frac{\pi}{2}$ radians. Sector B has radius 4 cm. Find the angle of Sector B in radians. Show all working. 3 MARKS
📖 Comprehensive answers (click to reveal)
Drill 1: $l = 6 \times \frac{\pi}{3} = 2\pi$ cm
Drill 2: $A = \frac{1}{2}(100) \times \frac{3\pi}{4} = \frac{75\pi}{2}$ cm$^2$
Drill 3: $45^\circ = \frac{\pi}{4}$ rad. $l = 12 \times \frac{\pi}{4} = 3\pi$ cm. $A = \frac{1}{2}(144) \times \frac{\pi}{4} = 18\pi$ cm$^2$.
Drill 4: $\theta = \frac{8\pi}{4} = 2\pi$ rad (a full circle)
Drill 5: $50\pi = \frac{1}{2}r^2 \times \frac{\pi}{2} \Rightarrow r^2 = 200 \Rightarrow r = 10\sqrt{2}$ cm
Q1 (3 marks): (a) $l = 9 \times \frac{2\pi}{3} = 6\pi$ cm [1.5]. (b) $A = \frac{1}{2}(81) \times \frac{2\pi}{3} = 27\pi$ cm$^2$ [1.5].
Q2 (3 marks): Perimeter $= 2r + l = 40$ [1]. So $l = 40 - 24 = 16$ cm [0.5]. Then $\theta = \frac{l}{r} = \frac{16}{12} = \frac{4}{3}$ rad [1.5].
Q3 (3 marks): Area of A $= \frac{1}{2}(36) \times \frac{\pi}{2} = 9\pi$ cm$^2$ [1]. Set equal to area of B: $9\pi = \frac{1}{2}(16) \times \theta_B$ [1]. So $\theta_B = \frac{9\pi}{8}$ rad [1].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering arc length and sector area questions. Lighter alternative to the boss.
Mark lesson as complete
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