088. Area of Rectangles and Metric Area Units

Learning Intentions

• To understand what the area of a two-dimensional shape is
• convert between metric areas (square millimetres, square centimetres, square metres, square kilometres, hectares)
• find the area of squares and other rectangles

Pre-requisite Summary

• Know that a two-dimensional shape is a flat shape with length and width
• Know that area measures the amount of surface covered inside a shape
• Know that area is measured in square units such as ${\text{mm}}^2$, ${\text{cm}}^2$, ${\text{m}}^2$, ${\text{km}}^2$
• Know that a square unit means a $1$ by $1$ square of that unit
• Know that length and width must be in the same unit before calculating area
• Know that converting area units is different from converting length units
• Know that $1\text{ cm}=10\text{ mm}$ but $1{\text{ cm}}^2=100{\text{ mm}}^2$
• Know that $1{\text{ m}}^2=10000{\text{ cm}}^2$
• Know that $1\text{ ha}=10000{\text{ m}}^2$
• Know that the area of a rectangle is found using $A=l\times w$

Worked Examples

Worked Example 1

Decide whether each statement describes area.
a) The space inside a shape
b) The distance around a shape
c) The amount of surface covered

Worked Example 2

Convert the following metric areas.
a) $6{\text{ cm}}^2$ to ${\text{mm}}^2$
b) $350{\text{ cm}}^2$ to ${\text{m}}^2$
c) $2.5{\text{ m}}^2$ to ${\text{cm}}^2$

Worked Example 3

Convert the following metric areas.
a) $3\text{ ha}$ to ${\text{m}}^2$
b) $45000{\text{ m}}^2$ to $\text{ha}$
c) $4{\text{ km}}^2$ to $\text{ha}$

Worked Example 4

Find the area of a square with side length $8\text{ cm}$.

Worked Example 5

Find the area of a rectangle with length $12\text{ m}$ and width $5\text{ m}$.

Worked Example 6

A rectangular playground is $18\text{ m}$ long and $700\text{ cm}$ wide. Convert to the same unit and find the area.

Problems

Problem 1

Decide whether each statement describes area.
a) The space inside a shape
b) The length of the boundary
c) The amount of surface covered

Problem 2

Convert the following metric areas.
a) $9{\text{ cm}}^2$ to ${\text{mm}}^2$
b) $500{\text{ cm}}^2$ to ${\text{m}}^2$
c) $1.8{\text{ m}}^2$ to ${\text{cm}}^2$

Problem 3

Convert the following metric areas.
a) $5\text{ ha}$ to ${\text{m}}^2$
b) $62000{\text{ m}}^2$ to $\text{ha}$
c) $2{\text{ km}}^2$ to $\text{ha}$

Problem 4

Find the area of a square with side length $11\text{ cm}$.

Problem 5

Find the area of a rectangle with length $15\text{ m}$ and width $6\text{ m}$.

Problem 6

A rectangular garden is $14\text{ m}$ long and $500\text{ cm}$ wide. Convert to the same unit and find the area.

Exercises

Understanding and Fluency

1. Decide whether each statement is about area.
   a) The amount of space inside a shape
   b) The distance around the outside of a shape
   c) The number of square units covering a surface

2. Choose the most suitable unit for area.
   a) The surface of a page
   b) The floor area of a classroom
   c) The area of a large farm

3. Convert the following.
   a) $4{\text{ cm}}^2$ to ${\text{mm}}^2$
   b) $12{\text{ cm}}^2$ to ${\text{mm}}^2$
   c) $25{\text{ cm}}^2$ to ${\text{mm}}^2$

4. Convert the following.
   a) $800{\text{ mm}}^2$ to ${\text{cm}}^2$
   b) $3{\text{ m}}^2$ to ${\text{cm}}^2$
   c) $20000{\text{ cm}}^2$ to ${\text{m}}^2$

5. Convert the following.
   a) $2\text{ ha}$ to ${\text{m}}^2$
   b) $70000{\text{ m}}^2$ to $\text{ha}$
   c) $1{\text{ km}}^2$ to $\text{ha}$

6. Find the area of each square.
   a) Side length $4\text{ cm}$
   b) Side length $9\text{ m}$
   c) Side length $12\text{ mm}$

7. Find the area of each rectangle.
   a) Length $7\text{ cm}$, width $3\text{ cm}$
   b) Length $15\text{ m}$, width $8\text{ m}$
   c) Length $2.5\text{ cm}$, width $6\text{ cm}$

8. Find the area of each rectangle after converting units where needed.
   a) $9\text{ m}$ by $300\text{ cm}$
   b) $120\text{ cm}$ by $4\text{ m}$
   c) $2\text{ m}$ by $250\text{ mm}$

Reasoning

9. Explain why area is measured in square units rather than ordinary length units.

10. A student says $1{\text{ cm}}^2=10{\text{ mm}}^2$ because $1\text{ cm}=10\text{ mm}$. Explain the error.

11. Explain why the formula $A=l\times w$ works for rectangles.

12. A student finds the area of a $6\text{ m}$ by $40\text{ cm}$ rectangle by doing $6\times 40=240{\text{ m}}^2$. Describe the mistake.

Problem-solving

13. A bedroom floor is $4\text{ m}$ long and $3.5\text{ m}$ wide. Find its area.

14. A square tile has side length $20\text{ cm}$. Find its area.

15. A rectangular sign is $120\text{ cm}$ long and $80\text{ cm}$ wide. Find its area in ${\text{cm}}^2$ and in ${\text{m}}^2$.

16. A farmer has a paddock with area $5\text{ ha}$. Write this area in ${\text{m}}^2$.

17. A park covers $2.4{\text{ km}}^2$. Write this area in hectares.

18. A rectangular vegetable patch is $15\text{ m}$ long and $8\text{ m}$ wide. How many square metres of soil does it cover?

Potential Misunderstandings

• A student may confuse area with perimeter and think area is the distance around a shape
• A student may forget that area describes the surface inside a two-dimensional shape
• A student may use length units such as cm or m instead of square units such as ${\text{cm}}^2$ or ${\text{m}}^2$
• A student may think converting area units uses the same scale factor as converting length units
• A student may think $1{\text{ cm}}^2=10{\text{ mm}}^2$ instead of $100{\text{ mm}}^2$
• A student may multiply side lengths that are in different units without converting first
• A student may add the side lengths of a rectangle instead of multiplying them
• A student may forget that a square is a rectangle with equal side lengths, so its area is still found by multiplying two side lengths
• A student may omit units or write incorrect units in the final answer

Next: 089. Area of Parallelograms