123. Area of Circles and Parts of Circles

Learning Intentions

• find the area of a circle given its radius or diameter using a calculator
• find the area of a circle given its radius or diameter using an approximation for $\pi$
• To understand how to find the area of a semicircle or quadrant by multiplying a circle’s area by ${\displaystyle \frac{1}{2}}$ or ${\displaystyle \frac{1}{4}}$

Pre-requisite Summary

• Know that area is the amount of surface inside a two-dimensional shape
• Know that area is measured in square units such as ${\text{cm}}^2$, ${\text{m}}^2$ and ${\text{mm}}^2$
• Know the meaning of radius and diameter in a circle
• Understand that the diameter is twice the radius
• Be able to substitute a number into a formula
• Be able to use a calculator to square a number
• Understand that an approximation is close to the exact value
• Know that a semicircle is half a circle and a quadrant is one quarter of a circle

Worked Examples

Worked Example 1

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

Worked Example 2

Find the area of each circle using a calculator:
a) radius $=4\text{ cm}$
b) diameter $=10\text{ cm}$
c) radius $=7.5\text{ m}$

Worked Example 3

Find the area of each circle using $\pi \approx 3.14$:
a) radius $=6\text{ cm}$
b) diameter $=14\text{ cm}$
c) radius $=3\text{ mm}$

Worked Example 4

A circle has diameter $18\text{ cm}$.
a) Find the radius
b) Find the area using a calculator

Worked Example 5

Find the area of each part of a circle using $\pi \approx 3.14$:
a) a semicircle with radius $8\text{ cm}$
b) a quadrant with radius $12\text{ cm}$

Worked Example 6

A circle has radius $5\text{ m}$.
a) Find the area of the whole circle using a calculator
b) Find the area of the semicircle
c) Find the area of the quadrant

Problems

Problem 1

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

Problem 2

Find the area of each circle using a calculator:
a) radius $=3\text{ cm}$
b) diameter $=12\text{ cm}$
c) radius $=6.2\text{ m}$

Problem 3

Find the area of each circle using $\pi \approx 3.14$:
a) radius $=5\text{ cm}$
b) diameter $=16\text{ cm}$
c) radius $=4\text{ mm}$

Problem 4

A circle has diameter $20\text{ cm}$.
a) Find the radius
b) Find the area using a calculator

Problem 5

Find the area of each part of a circle using $\pi \approx 3.14$:
a) a semicircle with radius $6\text{ cm}$
b) a quadrant with radius $10\text{ cm}$

Problem 6

A circle has radius $9\text{ m}$.
a) Find the area of the whole circle using a calculator
b) Find the area of the semicircle
c) Find the area of the quadrant

Exercises

Understanding and Fluency

1. Complete each statement:
   a) The area of a circle is the amount of ______ inside the circle
   b) The formula for the area of a circle is $A=\underset{―}{}$
   c) The diameter is ______ times the radius
   d) Area is measured in ______ units

2. Find the missing measure:
   a) radius $=7\text{ cm}$, diameter $=$ ?
   b) diameter $=18\text{ cm}$, radius $=$ ?
   c) radius $=2.5\text{ m}$, diameter $=$ ?
   d) diameter $=11\text{ mm}$, radius $=$ ?

3. Find the area of each circle using a calculator:
   a) radius $=2\text{ cm}$
   b) radius $=9\text{ cm}$
   c) diameter $=8\text{ cm}$

4. Find the area of each circle using $\pi \approx 3.14$:
   a) radius $=4\text{ cm}$
   b) diameter $=20\text{ cm}$
   c) radius $=11\text{ mm}$

5. Find the area of each circle:
   a) diameter $=14\text{ cm}$, using a calculator
   b) diameter $=14\text{ cm}$, using $\pi \approx 3.14$
   c) radius $=6.5\text{ m}$, using a calculator

6. Find the area of each part of a circle using $\pi \approx 3.14$:
   a) a semicircle with radius $7\text{ cm}$
   b) a quadrant with radius $8\text{ cm}$
   c) a semicircle with diameter $10\text{ cm}$

7. Find the area of each part of a circle:
   a) a quadrant with diameter $12\text{ cm}$
   b) a semicircle with radius $3.5\text{ m}$
   c) a quadrant with radius $9\text{ mm}$

8. Solve each:
   a) A circle has radius $10\text{ cm}$. Find the area of the whole circle and the semicircle using $\pi \approx 3.14$
   b) A circle has diameter $24\text{ cm}$. Find the area of the whole circle and the quadrant using $\pi \approx 3.14$
   c) A circle has radius $4\text{ m}$. Find the area of the whole circle using a calculator

Reasoning

9. Explain why the formula for the area of a circle uses $r^2$ and not just $r$.

10. A student says that if the diameter is $12\text{ cm}$, then the area is $\pi \times {12}^2$. Explain the mistake.

11. Noah says that a semicircle has area equal to $2\times \pi r^2$. Is he correct? Explain.

12. Explain why the area of a quadrant is found by multiplying the area of the whole circle by ${\displaystyle \frac{1}{4}}$.

13. A student finds the area of a circle and writes the answer in cm instead of ${\text{cm}}^2$. Describe the error.

Problem-solving

14. A circular garden has radius $5\text{ m}$. Find its area using $\pi \approx 3.14$.

15. A clock face has diameter $30\text{ cm}$. Find its area using a calculator.

16. A semicircular window has radius $8\text{ cm}$. Find its area using $\pi \approx 3.14$.

17. A quadrant of a circle has radius $12\text{ cm}$. Find its area using $\pi \approx 3.14$.

18. A circular pond has diameter $18\text{ m}$. Find the area of the pond using $\pi \approx 3.14$.

19. A pizza has radius $14\text{ cm}$. One quarter of the pizza is eaten. Find the area of the part that was eaten using $\pi \approx 3.14$.

Potential Misunderstandings

• Students may confuse area with circumference
• Students may confuse radius and diameter
• Students may forget to halve the diameter to find the radius before using the area formula
• Students may use $A=\pi d$ instead of $A=\pi r^2$ when calculating area.
• Students may forget to square the radius
• Students may square the diameter instead of the radius
• Students may round too early when using a calculator
• Students may forget that a semicircle is half of the whole circle
• Students may forget that a quadrant is one quarter of the whole circle
• Students may use linear units instead of square units in the final answer