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
• Know that the arc length is the same fraction of the circumference as the central angle is of
• Know the circumference formulas and
• 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 and a sector angle of . Solve the length of the arc.

Worked Example 3

A circle has radius and a sector angle of . Find the length of the arc.

Worked Example 4

A circle has diameter and a sector angle of . Find the length of the arc.

Worked Example 5

A sector has radius and angle . Find its perimeter.

Worked Example 6

A sector has radius and angle . 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 and a sector angle of . Find the length of the arc.

Problem 3

A circle has radius and a sector angle of . Find the length of the arc.

Problem 4

A circle has diameter and a sector angle of . Find the length of the arc.

Problem 5

A sector has radius and angle . Find its perimeter.

Problem 6

A sector has radius and angle . Find its perimeter.

Exercises

Understanding and Fluency

Exercise 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

Exercise 2.

Complete the statements.

a) A full circle has angle

b) An arc for is of the circumference

c) An arc for is of the circumference

Exercise 3.

A circle has circumference . Find the arc length for:

a)

b)

c)

Exercise 4.

Find the arc length.

a)

b)

c)

Exercise 5.

Find the arc length.

a)

b)

c)

Exercise 6.

Find the perimeter of each sector.

a)

b)

c)

Exercise 7.

Find the perimeter of each sector.

a)

b)

c)

Exercise 8.

A circle has circumference . Find:

a) the arc length for

b) the arc length for

c) the arc length for

Reasoning

Exercise 9.

Explain why the arc length for a sector is half the circumference.

Exercise 10.

A student says the arc length for a sector is found by multiplying the circumference by . Explain the error.

Exercise 11.

Two sectors are cut from circles of different sizes, but both have angle . Explain why their arc lengths are not necessarily equal.

Exercise 12.

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

Problem-solving

Exercise 13.

A pizza is cut into equal slices. The pizza has radius . Find the length of the crust on one slice.

Exercise 14.

A circular garden has radius . A sector with angle is used for a flower bed. Find the arc length of the flower bed edge.

Exercise 15.

A sector of a circle has radius and angle . Find its perimeter.

Exercise 16.

A clock face has radius . Find the length of the arc from to .

Exercise 17.

A sector-shaped path has radius and angle . Find the total distance around the outside of the path.

Exercise 18.

A circle has diameter . A sector contained within has a central angle of .

a) Find the arc length and

b) 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
• 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