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.
Today's hook, A pizza has radius 20 cm. A slice is cut with an angle of 45° at the centre. Without using any formulas, estimate the arc length of the crust on this slice.
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Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
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 $2\pi r$.
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The two moves
Work through the core explanation before applying it.
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The two moves
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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.
$l = r\theta$ $A = \tfrac{1}{2}r^2\theta$
Arc length
$l = r\theta$ when $\theta$ is in radians. This follows directly from the definition of a radian: $\theta = \frac{l}{r}$.
Sector area
$A = \frac{1}{2}r^2\theta$ when $\theta$ is in radians. Derived from taking the fraction $\frac{\theta}{2\pi}$ of the full circle area.
Convert first
If $\theta$ is given in degrees, convert to radians using $\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$ before substituting.
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The area of a segment
core concept
We just saw the arc length $l = r\theta$ and the sector area $A = \tfrac{1}{2}r^2\theta$. That raises a question: a sector is the whole slice, but what if you want only the part cut off by the straight chord? This card answers it → that region is a segment, and its area is the sector minus the triangle.
A segment is the region between a chord and the arc it cuts off. Draw the two radii to the ends of the chord and you have a sector, made up of the segment plus a triangle.
So the segment area is the sector area minus the triangle area. The sector is $\tfrac{1}{2}r^2\theta$, and the triangle has two sides $r$ with the included angle $\theta$, so its area is $\tfrac{1}{2}r^2\sin\theta$.
Subtracting gives $A = \tfrac{1}{2}r^2\theta - \tfrac{1}{2}r^2\sin\theta = \tfrac{1}{2}r^2(\theta - \sin\theta)$, with $\theta$ in radians.
Radians in both terms. The $\theta$ in $(\theta - \sin\theta)$ is a radian measure, and $\sin\theta$ must be evaluated with the calculator in radian mode. Mixing degrees and radians here is the standard way to lose the marks.
Quick check: the area of a minor segment is the area of the sector minus the area of what?
A segment is the region between a chord and its arc. Segment area $= $ sector $-$ triangle $= \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta = \frac{1}{2}r^2(\theta - \sin\theta)$, with $\theta$ in radians and the calculator in radian mode. A major segment is the circle minus the minor segment.
Pause, copy the segment diagram (chord, arc, two radii), the derivation as sector minus triangle, the formula $\frac{1}{2}r^2(\theta - \sin\theta)$, and the radian-mode warning, into your book.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
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
Understand
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
Can do
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
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Key terms
Arc lengthThe distance along the curved part of a sector: $l = r\theta$ (radians).
Sector areaThe area of a "pizza slice" region: $A = \frac{1}{2}r^2\theta$ (radians).
Central angleThe angle subtended at the centre of the circle by the sector.
Minor arcThe shorter of the two arcs between two points on a circle.
Major arcThe longer of the two arcs between two points on a circle.
Chord lengthThe straight-line distance between the endpoints of an arc: $2r\sin\frac{\theta}{2}$.
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Arc length and sector area, what's actually going on
Work through the core explanation before applying it.
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Arc length and sector area, what's actually going on
core concept
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:
$$l = r\theta$$
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:
$$A = \frac{1}{2}r^2\theta$$
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: $r = \sqrt{\frac{2A}{\theta}}$
Why satellite dishes are parabolic (and how arc length matters). Satellite dishes collect signals and focus them to a single point using a parabolic shape. But the manufacturing process often starts with a flat sheet of metal that is cut into sectors and bent into a cone, then flattened into a parabola. Engineers must calculate the arc length of each sector precisely so the metal pieces fit together perfectly. An error of even a few millimetres in arc length can defocus the signal and reduce reception quality.
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.
Complete: Arc length $l = $ where $\theta$ must be in .
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · ARC LENGTH
Find the arc length of a sector with radius 8 cm and central angle $\frac{\pi}{4}$ radians.
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$l = r\theta$
Identify the arc length formula.
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$l = 8 \times \frac{\pi}{4}$
Substitute $r = 8$ and $\theta = \frac{\pi}{4}$.
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$l = 2\pi$ cm
Simplify. $8 \div 4 = 2$.
PROBLEM 2 · SECTOR AREA
A sector has radius 10 cm and central angle $60^\circ$. Find its area.
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$60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3}$ rad
A sector has area $27\pi$ cm$^2$ and central angle $\frac{\pi}{2}$ radians. Find the radius.
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$A = \frac{1}{2}r^2\theta$
Start with the area formula.
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$27\pi = \frac{1}{2}r^2 \times \frac{\pi}{2}$
Substitute known values.
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$27\pi = \frac{\pi r^2}{4} \Rightarrow r^2 = 108 \Rightarrow r = \sqrt{108} = 6\sqrt{3}$ cm
Cancel $\pi$, multiply both sides by 4, then take the square root.
Quick check: A sector has $r = 6$ cm and $\theta = \frac{\pi}{3}$ rad. What is the arc length?
Common errors · the 3 traps that cost marks
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Dodge the traps, then apply
Meet the mistakes that cost marks, then do it yourself.
Trap 01
Using degrees directly in the radian formulas
The formulas $l = r\theta$ and $A = \frac{1}{2}r^2\theta$ ONLY work when $\theta$ is in radians. If you substitute $60$ (degrees) directly, your answer will be wrong by a factor of $\frac{\pi}{180}$. Always convert first.
Trap 02
Confusing arc length with chord length
The arc length is the curved distance along the circumference. The chord length is the straight-line distance between the two endpoints of the arc. These are different quantities with different formulas.
Trap 03
Forgetting the half in sector area
Some students write $A = r^2\theta$ instead of $A = \frac{1}{2}r^2\theta$, giving answers that are double the correct value. The $\frac{1}{2}$ appears because a sector is essentially a "curved triangle."
True or false: The formula $A = \frac{1}{2}r^2\theta$ gives the correct sector area when $\theta$ is in degrees.
Quick-fire practice · 5 reps
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Drill it, then lock it in
Run the quick drill and copy the summary into your book.
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$r = 6$ cm, $\theta = \frac{\pi}{3}$ rad. Find $l$.
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$r = 10$ cm, $\theta = \frac{3\pi}{4}$ rad. Find $A$.
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$r = 12$ cm, $\theta = 45^\circ$. Find $l$ and $A$.
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$l = 8\pi$ cm, $r = 4$ cm. Find $\theta$ in radians.
Match each formula to its purpose: Which formula finds the angle if you know arc length and radius?
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Revisit your thinking
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.
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In 10 words or fewer, explain why $A = \frac{1}{2}r^2\theta$ has a $\frac{1}{2}$ in it.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Write full responses, then check them against the model answers.
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Short answer
ApplyBand 4
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
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ApplyBand 4
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
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AnalyseBand 5
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
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].
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Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
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Take the full module quiz
quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
