GM Lesson 025 Circumference and Sector Arc Length

Learning Intentions

By the end of this lesson, students will be able to:

• Calculate the circumference of a circle using .
• Calculate arc length for sectors using the central angle and radius.
• Solve practical perimeter problems involving circles and sectors.

Prerequisites

Students should already be able to:

• Identify the radius and diameter of a circle.
• Substitute values into a formula.
• Use a calculator with .
• Calculate perimeters of standard two-dimensional shapes.
• Round answers to a stated number of decimal places.

Key Idea Summary

A circle has radius , diameter , and circumference .

The diameter is twice the radius:

The circumference of a circle is:

A sector is part of a circle formed by two radii and an arc.

The arc length of a sector is the same fraction of the circumference as its central angle is of :

where is the central angle in degrees and is the radius.

For a sector, the perimeter includes the arc length and the two radii:

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

Begin by reviewing that perimeter means the distance around the outside of a shape.

For a circle, the perimeter is called the circumference.

Explain that is the constant ratio between the circumference of a circle and its diameter. In this lesson, students should use the calculator button unless told otherwise.

The new CASIO 8200-AU needs to have it’s answers turned into decimal using the format key.

For sectors, emphasise the following reasoning:

If the central angle is , the arc length is the full circumference.

If the central angle is , the arc length is half the circumference.

If the central angle is , the arc length is one quarter of the circumference.

Therefore, for any central angle :

Worked Example 1: Circumference from Radius

Find the circumference of a circle with radius cm. Round to decimal place.

The circumference is approximately cm.

Worked Example 2: Circumference from Diameter

A circular table has diameter m. Find its circumference. Round to decimal places.

Since the diameter is m:

Now use the circumference formula:

The circumference is approximately m.

Worked Example 3: Arc Length of a Sector

Find the arc length of a sector with radius cm and central angle . Round to decimal place.

The arc length is approximately cm.

Worked Example 4: Perimeter of a Sector

A sector has radius cm and central angle . Find the perimeter of the sector. Round to decimal place.

First calculate the arc length:

The perimeter of the sector includes the arc length and two radii:

The perimeter is approximately cm.

Worked Example 5: Practical Perimeter Problem

A garden bed is shaped like a semicircle attached to a rectangle. The rectangle is m long and m wide. The semicircle is attached along one of the m sides. Find the outside perimeter of the garden bed. Round to decimal place.

The diameter of the semicircle is m, so:

The arc length of the semicircle is half the circumference:

The outside perimeter includes two m sides, one m side, and the semicircular arc:

The outside perimeter is approximately m.

Understanding Checks

Check 1

A circle has radius cm.

What formula should be used to calculate its circumference?

Expected response:

Check 2

A circle has diameter cm.

What is its radius?

Expected response:

Check 3

A sector has central angle .

What fraction of the full circumference is its arc length?

Expected response:

Check 4

A sector has radius cm and central angle .

Which expression gives the arc length?

Expected response:

Check 5

Why is the perimeter of a sector not just its arc length?

Expected response:

The perimeter includes the curved arc and the two straight radii.

Exercises

Simple Familiar Exercises

Exercise 1

Find the circumference of a circle with radius cm. Round to decimal place.

Exercise 2

Find the circumference of a circle with radius m. Round to decimal place.

Exercise 3

Find the circumference of a circle with diameter cm. Round to decimal place.

Exercise 4

Find the circumference of a circle with diameter m. Round to decimal places.

Exercise 5

Find the arc length of a sector with radius cm and central angle . Round to decimal place.

Exercise 6

Find the arc length of a sector with radius m and central angle . Round to decimal place.

Complex Familiar Exercises

Exercise 7

Find the arc length of a sector with radius cm and central angle . Round to decimal place.

Exercise 8

Find the arc length of a sector with radius m and central angle . Round to decimal places.

Exercise 9

A sector has radius cm and central angle .

Find its perimeter. Round to decimal place.

Exercise 10

A sector has radius cm and central angle .

Find its perimeter. Round to decimal place.

Exercise 11

A circular running track has radius m.

Find the distance around one full lap. Round to the nearest metre.

Exercise 12

A circular pond has diameter m.

A gardener wants to place edging around the pond. Find the length of edging required. Round to decimal place.

Homework Problems

Homework Problem 1

Find the circumference of a circle with radius cm. Round to decimal place.

Homework Problem 2

Find the circumference of a circle with diameter m. Round to decimal places.

Homework Problem 3

Find the arc length of a sector with radius cm and central angle . Round to decimal place.

Homework Problem 4

Find the arc length of a sector with radius m and central angle . Round to decimal places.

Homework Problem 5

A sector has radius cm and central angle .

Find the perimeter of the sector. Round to decimal place.

Homework Problem 6

A circular garden has diameter m.

A border is placed around the outside. Find the length of the border required. Round to decimal place.

Homework Problem 7

A semicircular arch has diameter m.

Find the length of the curved part of the arch. Round to decimal places.

Homework Problem 8

A running track bend is a sector with radius m and central angle .

Find the distance around the bend. Round to the nearest metre.

Next: GM Lesson 026 Areas of Polygons