086. Circles: Radius, Diameter and Circumference

Learning Intentions

• To know the features of a circle including the radius, diameter and circumference
• To know that $\pi$ is the ratio of the circumference of a circle to its diameter
• calculate a circle’s circumference, diameter or radius if given one of the other two measurements

Pre-requisite Summary

• Know that a circle is a two-dimensional shape with all points the same distance from its centre
• Know that a radius is a line from the centre of a circle to its edge
• Know that a diameter is a line across the circle through the centre
• Know that the diameter is twice the radius, so $d=2r$
• Know that circumference means the distance around the outside of a circle
• Know how to substitute values into a formula
• Know how to rearrange simple equations using inverse operations
• Know that $\pi$ is a constant and is approximately $3.14$

Worked Examples

Worked Example 1

Identify each feature in a circle diagram.
a) Radius
b) Diameter
c) Circumference

Worked Example 2

The radius of a circle is $6\text{ cm}$. Find:
a) the diameter
b) the circumference

Worked Example 3

The diameter of a circle is $14\text{ m}$. Find:
a) the radius
b) the circumference

Worked Example 4

The circumference of a circle is $31.4\text{ cm}$. Find:
a) the diameter
b) the radius

Worked Example 5

The circumference of a circle is $44\text{ cm}$. Use $C=\pi d$ to find the diameter in terms of $\pi$, then as a decimal.

Worked Example 6

Explain why $\pi ={\displaystyle \frac{C}{d}}$ for every circle, and use this rule to check whether a circle with circumference $25.12\text{ cm}$ and diameter $8\text{ cm}$ is consistent.

Problems

Problem 1

Identify each feature in a circle diagram.
a) Radius
b) Diameter
c) Circumference

Problem 2

The radius of a circle is $9\text{ cm}$. Find:
a) the diameter
b) the circumference

Problem 3

The diameter of a circle is $20\text{ m}$. Find:
a) the radius
b) the circumference

Problem 4

The circumference of a circle is $62.8\text{ cm}$. Find:
a) the diameter
b) the radius

Problem 5

The circumference of a circle is $30\text{ cm}$. Use $C=\pi d$ to find the diameter in terms of $\pi$, then as a decimal.

Problem 6

Use $\pi ={\displaystyle \frac{C}{d}}$ to check whether a circle with circumference $18.84\text{ m}$ and diameter $6\text{ m}$ is consistent.

Exercises

Understanding and Fluency

1. Name the circle feature described.
   a) The distance from the centre to the edge
   b) The distance across the circle through the centre
   c) The distance around the outside of the circle

2. Complete the relationships (Use the formulas)
   a) $d=\mathrm{\_}\mathrm{\_}\mathrm{\_}$
   b) $r=\mathrm{\_}\mathrm{\_}\mathrm{\_}$
   c) $C=\pi \mathrm{\_}\mathrm{\_}\mathrm{\_}$

3. Find the missing measurement.
   a) If $r=4\text{ cm}$, find $d$
   b) If $d=18\text{ cm}$, find $r$
   c) If $r=12\text{ mm}$, find $d$

4. Find the circumference.
   a) $r=5\text{ cm}$
   b) $d=10\text{ cm}$
   c) $r=7\text{ m}$

5. Find the circumference.
   a) $d=16\text{ cm}$
   b) $r=2.5\text{ cm}$
   c) $d=30\text{ mm}$

6. Find the diameter and radius.
   a) $C=18.84\text{ cm}$
   b) $C=62.8\text{ m}$
   c) $C=94.2\text{ mm}$

7. Use $\pi \approx 3.14$.
   a) Find $C$ when $d=9\text{ cm}$
   b) Find $d$ when $C=50.24\text{ cm}$
   c) Find $r$ when $C=43.96\text{ cm}$

8. Decide whether each statement is true or false.
   a) The diameter is twice the radius
   b) The circumference is the same as the diameter
   c) For any circle, ${\displaystyle \frac{C}{d}}=\pi$

Reasoning

9. Explain why the diameter is always twice the radius.

10. A student says $\pi$ is found by doing ${\displaystyle \frac{d}{C}}$. Explain the error.

11. A circle has diameter $12\text{ cm}$. One student uses $C=\pi r$ and another uses $C=\pi d$. Determine which method is correct and explain why.

12. Explain why two circles of different sizes still have the same value of ${\displaystyle \frac{C}{d}}$.

Problem-solving

13. A bike wheel has radius $35\text{ cm}$. Find its diameter and circumference.

14. A circular garden has diameter $8\text{ m}$. How far is it around the outside?

15. A round table has circumference $251.2\text{ cm}$. Find its diameter and radius.

16. A circular clock face has radius $14\text{ cm}$. Find the circumference.

17. A circular running track marker has circumference $37.68\text{ m}$. Find its diameter.

18. A student measures a jar lid and finds its diameter is $7\text{ cm}$. Estimate the circumference using $\pi \approx 3.14$.

Potential Misunderstandings

• A student may confuse radius and diameter
• A student may forget that the diameter goes through the centre of the circle
• A student may think circumference means the space inside the circle rather than the distance around it
• A student may use $d=r$ instead of $d=2r$
• A student may use $C=\pi r$ instead of $C=2\pi r$ or $C=\pi d$
• A student may think $\pi$ changes for different circles
• A student may reverse the ratio and write $\pi ={\displaystyle \frac{d}{C}}$
• A student may calculate the diameter correctly from the circumference but forget to halve it to find the radius
• A student may leave answers without units or use inconsistent units

Next: 087e. Arc Length and Sector Perimeter