087e. Arc Length and Sector Perimeter

Learning Intentions

• To know that an arc is a portion of a circle
• To understand how the length of an arc relates to the angle at the centre of the circle
• calculate the length of an arc
• calculate the perimeter of a sector

Pre-requisite Summary

• Know that the circumference is the distance around the whole circle
• Know that a sector is a part of a circle formed by two radii and an arc
• Know that the angle at the centre determines what fraction of the whole circle is being used
• Know that a full circle is ${360}^{\circ }$
• Know that the arc length is the same fraction of the circumference as the central angle is of ${360}^{\circ }$
• Know the circumference formulas $C=2\pi r$ and $C=\pi d$
• Know that the perimeter of a sector is made from two radii and the arc length
• Be able to substitute values into formulas and simplify

Worked Examples

Worked Example 1

A circle is divided by radii to form sectors. Identify:
a) the arc
b) the sector
c) the angle at the centre

Worked Example 2

A circle has radius $7\text{ cm}$ and a sector angle of ${90}^{\circ }$. Find the length of the arc.

Worked Example 3

A circle has radius $10\text{ m}$ and a sector angle of ${144}^{\circ }$. Find the length of the arc.

Worked Example 4

A circle has diameter $14\text{ cm}$ and a sector angle of ${120}^{\circ }$. Find the length of the arc.

Worked Example 5

A sector has radius $8\text{ cm}$ and angle ${60}^{\circ }$. Find its perimeter.

Worked Example 6

A sector has radius $12\text{ m}$ and angle ${150}^{\circ }$. Find its perimeter.

Problems

Problem 1

A circle is divided by radii to form sectors. Identify:
a) the arc
b) the sector
c) the angle at the centre

Problem 2

A circle has radius $14\text{ cm}$ and a sector angle of ${90}^{\circ }$. Find the length of the arc.

Problem 3

A circle has radius $9\text{ m}$ and a sector angle of ${160}^{\circ }$. Find the length of the arc.

Problem 4

A circle has diameter $20\text{ cm}$ and a sector angle of ${72}^{\circ }$. Find the length of the arc.

Problem 5

A sector has radius $6\text{ cm}$ and angle ${60}^{\circ }$. Find its perimeter.

Problem 6

A sector has radius $15\text{ m}$ and angle ${120}^{\circ }$. Find its perimeter.

Exercises

Understanding and Fluency

1. Name the circle part described.
   a) A portion of the circumference
   b) A region bounded by two radii and an arc
   c) The angle formed at the centre of the circle

2. Complete the statements.
   a) A full circle has angle $\mathrm{\_}\mathrm{\_}\mathrm{\_}$
   b) An arc for ${180}^{\circ }$ is $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ of the circumference
   c) An arc for ${90}^{\circ }$ is $\mathrm{\_}\mathrm{\_}\mathrm{\_}$ of the circumference

3. A circle has circumference $40\text{ cm}$. Find the arc length for:
   a) ${90}^{\circ }$
   b) ${180}^{\circ }$
   c) ${45}^{\circ }$

4. Find the arc length.
   a) $r=5\text{ cm},\theta ={72}^{\circ }$
   b) $r=7\text{ cm},\theta ={180}^{\circ }$
   c) $r=12\text{ cm},\theta ={30}^{\circ }$

5. Find the arc length.
   a) $d=10\text{ cm},\theta ={144}^{\circ }$
   b) $d=18\text{ m},\theta ={60}^{\circ }$
   c) $d=24\text{ mm},\theta ={300}^{\circ }$

6. Find the perimeter of each sector.
   a) $r=4\text{ cm},\theta ={90}^{\circ }$
   b) $r=10\text{ cm},\theta ={180}^{\circ }$
   c) $r=9\text{ m},\theta ={120}^{\circ }$

7. Find the perimeter of each sector.
   a) $r=6\text{ cm},\theta ={45}^{\circ }$
   b) $r=14\text{ cm},\theta ={270}^{\circ }$
   c) $r=8\text{ m},\theta ={135}^{\circ }$

8. A circle has circumference $62.8\text{ cm}$. Find:
   a) the arc length for ${180}^{\circ }$
   b) the arc length for ${90}^{\circ }$
   c) the arc length for ${36}^{\circ }$

Reasoning

9. Explain why the arc length for a ${180}^{\circ }$ sector is half the circumference.

10. A student says the arc length for a ${60}^{\circ }$ sector is found by multiplying the circumference by $60$. Explain the error.

11. Two sectors are cut from circles of different sizes, but both have angle ${90}^{\circ }$. Explain why their arc lengths are not necessarily equal.

12. A student finds the perimeter of a sector by using only the arc length. Explain what has been left out.

Problem-solving

13. A pizza is cut into $8$ equal slices. The pizza has radius $14\text{ cm}$. Find the length of the crust on one slice.

14. A circular garden has radius $6\text{ m}$. A sector with angle ${120}^{\circ }$ is used for a flower bed. Find the arc length of the flower bed edge.

15. A sector of a circle has radius $10\text{ cm}$ and angle ${72}^{\circ }$. Find its perimeter.

16. A clock face has radius $9\text{ cm}$. Find the length of the arc from $12$ to $3$.

17. A sector-shaped path has radius $20\text{ m}$ and angle ${150}^{\circ }$. Find the total distance around the outside of the path.

18. A circle has diameter $28\text{ cm}$. A sector has central angle ${45}^{\circ }$. Find the arc length and the perimeter of the sector.

Potential Misunderstandings

• A student may think an arc is a straight line joining two points on a circle
• A student may confuse an arc with a sector
• A student may not connect the central angle with the fraction of the whole circle
• A student may forget that the fraction used is ${\displaystyle \frac{\theta }{360}}$
• A student may use the radius as the arc length
• A student may use the whole circumference instead of just the required fraction
• A student may calculate the arc length correctly but forget to add the two radii for the perimeter of a sector
• A student may use the diameter in place of the radius incorrectly in the circumference formula
• A student may not simplify the answer correctly or may omit units

Next: 088. Area of Rectangles and Metric Area Units