Sec 3 E-Math · Arc length and sector area

Segment area questions with answers

Practice questions on segment area, 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

$AB$ is a chord of a circle with centre $O$ and radius $7$ cm, and angle $AOB = 1.8$ radians. Find the area of the shaded segment, correct to 3 significant figures.

Step 1 of 3

Area of the sector

\mathstrut \text{Sector} = \dfrac{1}{2} \times 7^2 \times 1.8

\text{Sector} = 44.1

Try a question: Segment area

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Questions

Question 1Segment area, angle in radians
$AB$ is a chord of a circle with centre $O$ and radius $4$ cm, and angle $AOB = 0.6$ radians. Find the area of the shaded segment, correct to 3 significant figures.
Show the worked solution
1. Step 1: Area of the sector $\mathstrut \text{Sector} = \dfrac{1}{2} \times 4^2 \times 0.6$ $\text{Sector} = 4.8$
2. Step 2: Area of triangle OAB (calculator in radians) $\text{Triangle} = \dfrac{1}{2} \times 4 \times 4 \times \sin 0.6$ $\text{Triangle} = 4.5171\ldots$
3. Step 3: Segment = sector − triangle $\mathstrut \text{Segment} = \text{Sector} - \text{Triangle}$ $\text{Segment} = 0.2828\ldots$ $\phantom{\text{Segment}} = 0.283$ cm² (3 s.f.)
Question 2Segment area, angle in radians
$AB$ is a chord of a circle with centre $O$ and radius $10$ cm, and angle $AOB = 2.4$ radians. Find the area of the shaded segment, correct to 3 significant figures.
Show the worked solution
1. Step 1: Area of the sector $\mathstrut \text{Sector} = \dfrac{1}{2} \times 10^2 \times 2.4$ $\text{Sector} = 120$
2. Step 2: Area of triangle OAB (calculator in radians) $\text{Triangle} = \dfrac{1}{2} \times 10 \times 10 \times \sin 2.4$ $\text{Triangle} = 33.7731\ldots$
3. Step 3: Segment = sector − triangle $\mathstrut \text{Segment} = \text{Sector} - \text{Triangle}$ $\text{Segment} = 86.2268\ldots$ $\phantom{\text{Segment}} = 86.2$ cm² (3 s.f.)
Question 3Segment area, angle in degrees
$AB$ is a chord of a circle with centre $O$ and radius $8$ cm, and angle $AOB = 60^\circ$ . Find the area of the shaded segment, correct to 3 significant figures.
Show the worked solution
1. Step 1: Area of the sector $\mathstrut \text{Sector} = \dfrac{60}{360} \times \pi(8)^2$ $\text{Sector} = 33.5103\ldots$
2. Step 2: Area of triangle OAB $\text{Triangle} = \dfrac{1}{2} \times 8 \times 8 \times \sin 60^\circ$ $\text{Triangle} = 27.7128\ldots$
3. Step 3: Segment = sector − triangle $\mathstrut \text{Segment} = \text{Sector} - \text{Triangle}$ $\text{Segment} = 5.7975\ldots$ $\phantom{\text{Segment}} = 5.80$ cm² (3 s.f.)
Question 4Segment area, angle in degrees
$AB$ is a chord of a circle with centre $O$ and radius $9$ cm, and angle $AOB = 135^\circ$ . Find the area of the shaded segment, correct to 3 significant figures.
Show the worked solution
1. Step 1: Area of the sector $\mathstrut \text{Sector} = \dfrac{135}{360} \times \pi(9)^2$ $\text{Sector} = 95.4258\ldots$
2. Step 2: Area of triangle OAB $\text{Triangle} = \dfrac{1}{2} \times 9 \times 9 \times \sin 135^\circ$ $\text{Triangle} = 28.6378\ldots$
3. Step 3: Segment = sector − triangle $\mathstrut \text{Segment} = \text{Sector} - \text{Triangle}$ $\text{Segment} = 66.7880\ldots$ $\phantom{\text{Segment}} = 66.8$ cm² (3 s.f.)
Question 5Perimeter of a segment
$AB$ is a chord of a circle with centre $O$ and radius $9$ cm, and angle $AOB = 1.2$ radians. Find the perimeter of the minor segment, correct to 3 significant figures.
Show the worked solution
1. Step 1: The arc: s = rθ $\mathstrut s = 9 \times 1.2$ $s = 10.8$
2. Step 2: The chord AB: which rule in triangle OAB? $\text{Cosine rule}$
3. Step 3: Write the rule with the values (calculator in radians) $AB^2 = 9^2 + 9^2 - 2(9)(9) \cos 1.2$ $AB^2 = 103.2980\ldots$ $AB = 10.1635\ldots$
4. Step 4: Perimeter of the segment = arc + chord $\text{Perimeter} = 20.9635\ldots$ $\phantom{\text{Perimeter}} = 21.0$ cm (3 s.f.)
Question 6Perimeter of a segment
$AB$ is a chord of a circle with centre $O$ and radius $9$ cm, and angle $AOB = 2.4$ radians. Find the perimeter of the minor segment, correct to 3 significant figures.
Show the worked solution
1. Step 1: The arc: s = rθ $\mathstrut s = 9 \times 2.4$ $s = 21.6$
2. Step 2: The chord AB: which rule in triangle OAB? $\text{Cosine rule}$
3. Step 3: Write the rule with the values (calculator in radians) $AB^2 = 9^2 + 9^2 - 2(9)(9) \cos 2.4$ $AB^2 = 281.4577\ldots$ $AB = 16.7767\ldots$
4. Step 4: Perimeter of the segment = arc + chord $\text{Perimeter} = 38.3767\ldots$ $\phantom{\text{Perimeter}} = 38.4$ cm (3 s.f.)

More on arc length and sector area

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

Endless questions on segment area

A free account gives you new questions every time, step-by-step help, and tracks what you've mastered.

Start free

Worked example

AB

is a chord of a circle with centre

O

and radius

7

cm, and angle

AOB = 1.8

radians. Find the area of the shaded segment, correct to 3 significant figures.

Step 1 of 3

Area of the sector

\mathstrut \text{Sector} = \dfrac{1}{2} \times 7^2 \times 1.8

\text{Sector} = 44.1

Questions

Question 1Segment area, angle in radians

$AB$ is a chord of a circle with centre $O$ and radius $4$ cm, and angle $AOB = 0.6$ radians. Find the area of the shaded segment, correct to 3 significant figures.

Show the worked solution

Step 1: Area of the sector $\mathstrut \text{Sector} = \dfrac{1}{2} \times 4^2 \times 0.6$ $\text{Sector} = 4.8$
Step 2: Area of triangle OAB (calculator in radians) $\text{Triangle} = \dfrac{1}{2} \times 4 \times 4 \times \sin 0.6$ $\text{Triangle} = 4.5171\ldots$
Step 3: Segment = sector − triangle $\mathstrut \text{Segment} = \text{Sector} - \text{Triangle}$ $\text{Segment} = 0.2828\ldots$ $\phantom{\text{Segment}} = 0.283$ cm² (3 s.f.)

Question 2Segment area, angle in radians

$AB$ is a chord of a circle with centre $O$ and radius $10$ cm, and angle $AOB = 2.4$ radians. Find the area of the shaded segment, correct to 3 significant figures.

Show the worked solution

Step 1: Area of the sector $\mathstrut \text{Sector} = \dfrac{1}{2} \times 10^2 \times 2.4$ $\text{Sector} = 120$
Step 2: Area of triangle OAB (calculator in radians) $\text{Triangle} = \dfrac{1}{2} \times 10 \times 10 \times \sin 2.4$ $\text{Triangle} = 33.7731\ldots$
Step 3: Segment = sector − triangle $\mathstrut \text{Segment} = \text{Sector} - \text{Triangle}$ $\text{Segment} = 86.2268\ldots$ $\phantom{\text{Segment}} = 86.2$ cm² (3 s.f.)

Question 3Segment area, angle in degrees

$AB$ is a chord of a circle with centre $O$ and radius $8$ cm, and angle $AOB = 60^\circ$ . Find the area of the shaded segment, correct to 3 significant figures.

Show the worked solution

Step 1: Area of the sector $\mathstrut \text{Sector} = \dfrac{60}{360} \times \pi(8)^2$ $\text{Sector} = 33.5103\ldots$
Step 2: Area of triangle OAB $\text{Triangle} = \dfrac{1}{2} \times 8 \times 8 \times \sin 60^\circ$ $\text{Triangle} = 27.7128\ldots$
Step 3: Segment = sector − triangle $\mathstrut \text{Segment} = \text{Sector} - \text{Triangle}$ $\text{Segment} = 5.7975\ldots$ $\phantom{\text{Segment}} = 5.80$ cm² (3 s.f.)

Question 4Segment area, angle in degrees

$AB$ is a chord of a circle with centre $O$ and radius $9$ cm, and angle $AOB = 135^\circ$ . Find the area of the shaded segment, correct to 3 significant figures.

Show the worked solution

Step 1: Area of the sector $\mathstrut \text{Sector} = \dfrac{135}{360} \times \pi(9)^2$ $\text{Sector} = 95.4258\ldots$
Step 2: Area of triangle OAB $\text{Triangle} = \dfrac{1}{2} \times 9 \times 9 \times \sin 135^\circ$ $\text{Triangle} = 28.6378\ldots$
Step 3: Segment = sector − triangle $\mathstrut \text{Segment} = \text{Sector} - \text{Triangle}$ $\text{Segment} = 66.7880\ldots$ $\phantom{\text{Segment}} = 66.8$ cm² (3 s.f.)

Question 5Perimeter of a segment

$AB$ is a chord of a circle with centre $O$ and radius $9$ cm, and angle $AOB = 1.2$ radians. Find the perimeter of the minor segment, correct to 3 significant figures.

Show the worked solution

Step 1: The arc: s = rθ $\mathstrut s = 9 \times 1.2$ $s = 10.8$
Step 2: The chord AB: which rule in triangle OAB? $\text{Cosine rule}$
Step 3: Write the rule with the values (calculator in radians) $AB^2 = 9^2 + 9^2 - 2(9)(9) \cos 1.2$ $AB^2 = 103.2980\ldots$ $AB = 10.1635\ldots$
Step 4: Perimeter of the segment = arc + chord $\text{Perimeter} = 20.9635\ldots$ $\phantom{\text{Perimeter}} = 21.0$ cm (3 s.f.)

Question 6Perimeter of a segment

$AB$ is a chord of a circle with centre $O$ and radius $9$ cm, and angle $AOB = 2.4$ radians. Find the perimeter of the minor segment, correct to 3 significant figures.

Show the worked solution

Step 1: The arc: s = rθ $\mathstrut s = 9 \times 2.4$ $s = 21.6$
Step 2: The chord AB: which rule in triangle OAB? $\text{Cosine rule}$
Step 3: Write the rule with the values (calculator in radians) $AB^2 = 9^2 + 9^2 - 2(9)(9) \cos 2.4$ $AB^2 = 281.4577\ldots$ $AB = 16.7767\ldots$
Step 4: Perimeter of the segment = arc + chord $\text{Perimeter} = 38.3767\ldots$ $\phantom{\text{Perimeter}} = 38.4$ cm (3 s.f.)