Arc Length and Area of a Sector
From the crust of a pizza slice to the area covered by a garden sprinkler, parts of circles appear everywhere. This guide shows you how to calculate the length of a curved edge (an arc) and the area of a “slice” (a sector) by thinking of them as fractions of a full circle.
What are Arcs and Sectors?
Arc
An arc is a part of the circumference of a circle — a curved line measuring length.
Sector
A sector is a part of the area of a circle — like a pizza slice, bounded by two radii and an arc.
How to Calculate Arcs and Sectors
Both formulas are simply a fraction of the full circle. The fraction is $\dfrac{\theta}{360}$, where $\theta$ is the central angle in degrees.
Arc Length
Fraction of the whole circumference:
Equivalently: $\dfrac{\theta}{360} \times \pi d$.
Area of a Sector
Fraction of the whole circle’s area:
Area is in square units (cm², m²). The $r^2$ reminds you of that.
Worked Examples
Example 1: Arc Length
Radius 7 cm, central angle 120°. Find the arc length.
- $\dfrac{120}{360} \times 2\pi \times 7$.
- $= \dfrac{1}{3} \times 14\pi \approx 14.7$.
Answer: 14.7 cm (1 d.p.)
Example 2: Area of a Sector
Radius 10 m, central angle 80°. Find the area.
- $\dfrac{80}{360} \times \pi \times 10^2$.
- $= \dfrac{2}{9} \times 100\pi \approx 69.8$.
Answer: 69.8 m² (3 s.f.)
Example 3: Working Backwards (Finding the Radius)
Arc length 12 cm, central angle 150°. Find the radius to 1 d.p.
- $12 = \dfrac{150}{360} \times 2\pi r$.
- $12 = \dfrac{5}{12} \times 2\pi r \implies 12 = \dfrac{10\pi r}{12}$.
- $144 = 10\pi r \implies r = \dfrac{144}{10\pi} \approx 4.583\ldots$
Answer: 4.6 cm (1 d.p.)
Tutor Insights
🤔 Common Misunderstandings
- Mixing up the formulas. Arc length uses $r$; sector area uses $r^2$. The $r^2$ signals “square units”.
- Using the diameter instead of the radius. Halve the diameter before substituting.
- Dividing by 180 instead of 360. A full circle is always 360°.
📝 Common Exam Mistakes
- Forgetting $\dfrac{\theta}{360}$ and computing the full circumference or area.
- Calculator errors with $\pi$ — work step by step and keep full precision until the final answer.
- Wrong units — length answers in cm/m; area answers in cm²/m².
Practice Questions
- Radius 5 cm, central angle 90°. Find the arc length to 1 decimal place.
- Radius 8 m, central angle 45°. Find the sector area to 3 significant figures.
- A circular garden has diameter 12 m. A path follows an arc of 60°. Find the path length to 2 decimal places.
Show Answers
- $\frac{90}{360}\times2\pi\times5=\frac{1}{4}\times10\pi=2.5\pi\approx\mathbf{7.9\text{ cm}}$.
- $\frac{45}{360}\times\pi\times64=\frac{1}{8}\times64\pi=8\pi\approx\mathbf{25.1\text{ m}^2}$.
- $r=6\text{ m}$. $\frac{60}{360}\times2\pi\times6=\frac{1}{6}\times12\pi=2\pi\approx\mathbf{6.28\text{ m}}$.
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