Sec 3 E-Math · Arc length and sector area

Arc length questions with answers

Practice questions on arc length, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

A sector of a circle has radius $9$ cm and angle $210^\circ$ . Find the length of its arc, correct to 3 significant figures.

Step 1 of 2

The arc is a fraction of the circumference

s = \dfrac{\theta}{360} \times 2\pi r

\mathstrut s = \dfrac{210}{360} \times 2\pi(9)

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Questions

Question 1Arc length, angle in degrees
A sector of a circle has radius $8$ cm and angle $80^\circ$ . Find the length of its arc, correct to 3 significant figures.
Show the worked solution
1. Step 1: The arc is a fraction of the circumference $s = \dfrac{\theta}{360} \times 2\pi r$ $\mathstrut s = \dfrac{80}{360} \times 2\pi(8)$
2. Step 2: Work it out $s = 11.1701\ldots$ $\phantom{s} = 11.2$ cm (3 s.f.)
Question 2Arc length, angle in radians
A sector of a circle has radius $7$ cm and angle $1.2$ radians. Find the length of its arc.
Show the worked solution
1. Step 1: In radians, arc length = rθ $s = r\theta$ $\mathstrut s = 7 \times 1.2$ $s = 8.4$ cm
Question 3Find the angle in radians
A sector of a circle of radius $5$ cm has an arc of length $14$ cm. Find the angle, in radians, of the sector.
Show the worked solution
1. Step 1: s = rθ, so θ = s ÷ r $14 = 5\theta$ $\mathstrut \theta = \dfrac{14}{5}$ $\theta = 2.8$ rad
Question 4Find the radius
A sector with angle $\dfrac{\pi}{3}$ radians has an arc of length $20$ cm. Find the radius, correct to 3 significant figures.
Show the worked solution
1. Step 1: s = rθ, so r = s ÷ θ $20 = r \times \dfrac{\pi}{3}$ $r = \dfrac{20}{\dfrac{\pi}{3}}$ $r = 19.0985\ldots$ $\phantom{r} = 19.1$ cm (3 s.f.)
Question 5Perimeter of a sector
A sector of a circle has radius $5$ cm and angle $\dfrac{2\pi}{3}$ radians. Find its perimeter, correct to 3 significant figures.
Show the worked solution
1. Step 1: First the arc: s = rθ $\mathstrut s = 5 \times \dfrac{2\pi}{3}$ $s = 10.4719\ldots$
2. Step 2: The perimeter is the arc plus two radii $\mathstrut \text{Perimeter} = 5 + 5 + s$ $\text{Perimeter} = 20.4719\ldots$ $\phantom{\text{Perimeter}} = 20.5$ cm (3 s.f.)

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Notes: Arc length and sector area Worksheet with answers (PDF)

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Worked example

A sector of a circle has radius

9

cm and angle

210^\circ

. Find the length of its arc, correct to 3 significant figures.

Step 1 of 2

The arc is a fraction of the circumference

s = \dfrac{\theta}{360} \times 2\pi r

\mathstrut s = \dfrac{210}{360} \times 2\pi(9)

Questions

Question 1Arc length, angle in degrees

A sector of a circle has radius $8$ cm and angle $80^\circ$ . Find the length of its arc, correct to 3 significant figures.

Show the worked solution

Step 1: The arc is a fraction of the circumference $s = \dfrac{\theta}{360} \times 2\pi r$ $\mathstrut s = \dfrac{80}{360} \times 2\pi(8)$
Step 2: Work it out $s = 11.1701\ldots$ $\phantom{s} = 11.2$ cm (3 s.f.)

Question 2Arc length, angle in radians

A sector of a circle has radius $7$ cm and angle $1.2$ radians. Find the length of its arc.

Show the worked solution

Step 1: In radians, arc length = rθ $s = r\theta$ $\mathstrut s = 7 \times 1.2$ $s = 8.4$ cm

Question 3Find the angle in radians

A sector of a circle of radius $5$ cm has an arc of length $14$ cm. Find the angle, in radians, of the sector.

Show the worked solution

Step 1: s = rθ, so θ = s ÷ r $14 = 5\theta$ $\mathstrut \theta = \dfrac{14}{5}$ $\theta = 2.8$ rad

Question 4Find the radius

A sector with angle $\dfrac{\pi}{3}$ radians has an arc of length $20$ cm. Find the radius, correct to 3 significant figures.

Show the worked solution

Step 1: s = rθ, so r = s ÷ θ $20 = r \times \dfrac{\pi}{3}$ $r = \dfrac{20}{\dfrac{\pi}{3}}$ $r = 19.0985\ldots$ $\phantom{r} = 19.1$ cm (3 s.f.)

Question 5Perimeter of a sector

A sector of a circle has radius $5$ cm and angle $\dfrac{2\pi}{3}$ radians. Find its perimeter, correct to 3 significant figures.

Show the worked solution

Step 1: First the arc: s = rθ $\mathstrut s = 5 \times \dfrac{2\pi}{3}$ $s = 10.4719\ldots$
Step 2: The perimeter is the arc plus two radii $\mathstrut \text{Perimeter} = 5 + 5 + s$ $\text{Perimeter} = 20.4719\ldots$ $\phantom{\text{Perimeter}} = 20.5$ cm (3 s.f.)