# Stage 4 - space and geometry – 2D space

## Strategy

Students can:

• Use a formula to find the circumference of a circle
• Identify that a fraction of a circle comprises a sector
• Calculate the perimeter of a figure
• Correctly round a calculation to a given unit

## Activities to support the strategy

Students need to have confidence in using a formula to find the circumference of a circle, understand that sector is a fraction of a circle and have a well-developed understanding of perimeter. Review of each of these areas is encouraged before they are combined in more challenging perimeter problems.

### Activity 1

1. Review the use of formula to find the circumference of a circle. Students need to apply either of the formulas below to calculate the circumference of a circle with confidence, given a particular radius or diameter. At this stage students should also have modelled concepts around rounding. It should be noted that π is an irrational number and answers can be given rounded to a given number of places or in exact form (in terms of π). Appropriate use of approximations of π and the calculator should also be discussed. Formulas and examples are shown below. Calculate the circumference, given radius = 6 cm.

• Use π on your calculator and answer to 1 decimal place.
• c=2πr
• c=2×π×6
• c=18.849555…
• ∴ Circumference is 18.8 cm (1 d.p.) Calculate the circumference, given diameter = 10 m. Leave your answer in exact form

• c=πd
• c=π×10
• c=10π
• ∴ Circumference is 10 π m

2. A number of examples demonstrating the perimeter of more complex figures involving circles and in particular the perimeter of a sector need to be presented. Students should also be given the opportunity to practice a variety of questions to build their understanding. Calculate the perimeter.

• Take π=3.14 .

The perimeter of this figure is made up of a half-circle and the base. Using formula c=πd

• P=12 ×3.14×4+4
• P=10.28
• ∴ Perimeter is 10.28 cm Calculate the perimeter.

• Leave answer in terms of π

The perimeter of this figure is made up of a quarter-circle plus the two straight sides. Using formula c=2πr

• P=14 ×(2×π×3)+3+3
• P=3π2 +6
• ∴ Perimeter is (3π2 +6)  cm Calculate the perimeter. Use π on your calculator and answer correct to 2 decimal places

The perimeter of this figure is the two straight sides plus the curved fraction of the circle. Since there are 360° in a full revolution, the curved part of the perimeter is 50360 ×2πr

• P=(50360 ×2×π×4)+4+4
• P=11.49065…
• ∴ Perimeter is 11.49 cm (2 d.p.)

### Activity 2

Teachers should review the TIMES teacher guide on circles. A number of examples and teaching techniques are provided for measuring the perimeter of circles and more complex figures involving circles.

### Activity 3

Students and teachers can review the activities provided on Supporting Australian Mathematics Project - Circles. The resource may be useful in helping students achieve that element of the year 8 achievement standards that refers to students naming the features of circles and calculating areas and circumferences of circles.

## References

### Australian curriculum

ACMMG197: Investigate the relationship between features of circles, such as the circumference, radius and diameter; use formulas to solve problems involving circumference

### NSW syllabus

MA4-12MG: Calculates the perimeters of plane shapes and the circumferences of circles