120r. Review Circles and Circumference

Learning Intentions

• To know the meaning of the terms diameter, radius and circumference
• To understand that pi ($\pi$) is a number that equals the circumference of any circle divided by its diameter
• find the circumference of a circle using a calculator
• find the circumference of a circle using an approximation for $\pi$

Pre-requisite Summary

• Know that a circle is a closed shape where all points on the boundary are the same distance from the centre
• Understand that length is measured in one-dimensional units such as mm, cm and m
• Be able to multiply whole numbers and decimals
• Know how to use a calculator for multiplication
• Understand that a formula is a rule for calculating a quantity
• Be able to substitute a given number into a formula
• Know that an approximation is a value close to the exact value

Worked Examples

Worked Example 1

State the meaning of each term:
a) radius
b) diameter
c) circumference

Worked Example 2

Use circle facts to find the missing measure:
a) radius $=6$ cm. Find the diameter
b) diameter $=14$ mm. Find the radius
c) radius $=3.5$ m. Find the diameter

Worked Example 3

Use a calculator to find the circumference:
a) diameter $=8$ cm
b) radius $=5$ cm
c) diameter $=12.4$ m

Worked Example 4

Use $\pi \approx 3.14$ to find the circumference:
a) diameter $=9$ cm
b) radius $=7$ cm
c) diameter $=15$ mm

Worked Example 5

A circle has circumference $31.4$ cm using $\pi \approx 3.14$. Find the diameter.

Worked Example 6

A circle has radius $4$ cm.
a) Find the circumference using a calculator
b) Find the circumference using $\pi \approx 3.14$

Problems

Problem 1

State the meaning of each term:
a) radius
b) diameter
c) circumference

Problem 2

Use circle facts to find the missing measure:
a) radius $=9$ cm. Find the diameter
b) diameter $=18$ mm. Find the radius
c) radius $=2.5$ m. Find the diameter

Problem 3

Use a calculator to find the circumference:
a) diameter $=10$ cm
b) radius $=6$ cm
c) diameter $=7.2$ m

Problem 4

Use $\pi \approx 3.14$ to find the circumference:
a) diameter $=11$ cm
b) radius $=4$ cm
c) diameter $=20$ mm

Problem 5

A circle has circumference $62.8$ cm using $\pi \approx 3.14$. Find the diameter.

Problem 6

A circle has radius $8$ cm.
a) Find the circumference using a calculator
b) Find the circumference using $\pi \approx 3.14$

Exercises

Understanding and Fluency

1. State the meaning of each term:
   a) radius
   b) diameter
   c) circumference
   d) centre

2. Complete each statement:
   a) The diameter is ______ times the radius
   b) The radius is ______ the diameter
   c) The circumference is the distance ______ a circle

3. Find the missing measure:
   a) radius $=5$ cm, diameter $=$ ?
   b) diameter $=16$ cm, radius $=$ ?
   c) radius $=12$ mm, diameter $=$
   d) diameter $=9$ m, radius $=$

4. Use a calculator to find the circumference:
   a) diameter $=6$ cm
   b) radius $=3$ cm
   c) diameter $=14$ cm

5. Use $\pi \approx 3.14$ to find the circumference:
   a) diameter $=8$ cm
   b) radius $=10$ cm
   c) diameter $=25$ mm

6. Find the circumference of each circle:
   a) radius $=2.5$ cm, using a calculator
   b) radius $=2.5$ cm, using $\pi \approx 3.14$
   c) diameter $=18$ m, using $\pi \approx 3.14$

7. Use $\pi \approx 3.14$ to find the missing measure:
   a) circumference $=31.4$ cm, diameter $=$ ?
   b) circumference $=18.84$ cm, radius $=$ ?
   c) circumference $=62.8$ mm, diameter $=$

8. Choose the correct formula and evaluate:
   a) circumference when the diameter is known
   b) circumference when the radius is known
   c) diameter when the radius is known

Reasoning

9. Explain why the diameter of a circle is always twice the radius.

10. A student says that $\pi$ is different for different circles because large circles have larger circumferences. Explain the mistake.

11. Noah says that circumference is measured in square centimetres because a circle is a two-dimensional shape. Is he correct? Explain.

12. Explain why $C=\pi d$ and $C=2\pi r$ are equivalent formulas.

13. A student uses $C=\pi r$ to find the circumference. Describe the error.

Problem-solving

14. A bicycle wheel has diameter $70$ cm. Find its circumference using $\pi \approx 3.14$.

15. A circular table has radius $0.8$ m. Find its circumference using a calculator.

16. A hoop has circumference $94.2$ cm using $\pi \approx 3.14$. Find its diameter.

17. A circular pond has radius $6$ m. Find the circumference using $\pi \approx 3.14$.

18. A clock face has diameter $30$ cm. Find the circumference using both a calculator and $\pi \approx 3.14$.

19. A circle has circumference $125.6$ mm using $\pi \approx 3.14$. Find its radius.

Potential Misunderstandings

• Students may confuse radius and diameter
• Students may forget that the diameter is twice the radius
• Students may think circumference means the space inside the circle rather than the distance around it
• Students may think $\pi$ is measured in units rather than being a number
• Students may think $\pi$ changes from one circle to another
• Students may use the wrong formula, such as $C=\pi r$ instead of $C=2\pi r$ or $C=\pi d$
• Students may forget to double the radius before using $C=\pi d$
• Students may use square units for circumference instead of linear units
• Students may round too early when using a calculator
• Students may confuse using an exact calculator value of $\pi$ with using an approximation such as $3.14$