121r. Review Area of Shapes and Composite Figures

Learning Intentions

• To understand what the area of a two-dimensional shape is
• convert between different metric units of area, including hectares
• find the area of squares, rectangles, parallelograms and triangles
• To understand that the area of composite shapes can be found by adding or subtracting the area of more basic shapes

Pre-requisite Summary

• Know that two-dimensional shapes are flat shapes with length and width
• Understand that area measures the amount of surface inside a shape
• Know that area is measured in square units such as ${\text{mm}}^2$, ${\text{cm}}^2$, ${\text{m}}^2$ and ${\text{km}}^2$
• Know basic metric length conversions such as $1\text{ m}=100\text{ cm}$ and $1\text{ km}=1000\text{ m}$
• Be able to multiply whole numbers and decimals
• Understand that a square has all sides equal and a rectangle has opposite sides equal
• Know that a triangle can be seen as half of a matching parallelogram or rectangle
• Be able to split a complex shape into simpler shapes

Worked Examples

Worked Example 1

State what area means and identify suitable units:

a) the space inside a book cover
b) the space inside a classroom floor
c) the space inside a farm

Worked Example 2

Convert between metric units of area:

a) $3{\text{ m}}^2$ to ${\text{cm}}^2$
b) $45000{\text{ cm}}^2$ to ${\text{m}}^2$
c) $2\text{ ha}$ to ${\text{m}}^2$

Worked Example 3

Find the area:

a) a square with side length $6\text{ cm}$
b) a rectangle with length $9\text{ cm}$ and width $4\text{ cm}$
c) a parallelogram with base $8\text{ m}$ and perpendicular height $5\text{ m}$

Worked Example 4

Find the area of each triangle:

a) base $10\text{ cm}$, height $7\text{ cm}$
b) base $12\text{ m}$, height $4\text{ m}$

Worked Example 5

Find the area of a composite shape by addition:

a) a shape made from a $6\text{ cm}\times 4\text{ cm}$ rectangle and a $3\text{ cm}\times 2\text{ cm}$ rectangle
b) a shape made from two rectangles of areas $18{\text{ cm}}^2$ and $12{\text{ cm}}^2$

Worked Example 6

Find the area of a composite shape by subtraction:

a) a $10\text{ cm}\times 8\text{ cm}$ rectangle with a $4\text{ cm}\times 3\text{ cm}$ rectangle removed
b) a square of side $9\text{ m}$ with a triangle of area $12{\text{ m}}^2$ removed

Problems

Problem 1

State what area means and identify suitable units:

a) the space inside a desktop
b) the space inside a sports field
c) the space inside a state park

Problem 2

Convert between metric units of area:

a) $5{\text{ m}}^2$ to ${\text{cm}}^2$
b) $32000{\text{ cm}}^2$ to ${\text{m}}^2$
c) $4\text{ ha}$ to ${\text{m}}^2$

Problem 3

Find the area:

a) a square with side length $7\text{ cm}$
b) a rectangle with length $11\text{ cm}$ and width $5\text{ cm}$
c) a parallelogram with base $9\text{ m}$ and perpendicular height $6\text{ m}$

Problem 4

Find the area of each triangle:

a) base $8\text{ cm}$, height $9\text{ cm}$
b) base $15\text{ m}$, height $6\text{ m}$

Problem 5

Find the area of a composite shape by addition:

a) a shape made from a $7\text{ cm}\times 3\text{ cm}$ rectangle and a $4\text{ cm}\times 2\text{ cm}$ rectangle
b) a shape made from two rectangles of areas $24{\text{ cm}}^2$ and $15{\text{ cm}}^2$

Problem 6

Find the area of a composite shape by subtraction:

a) a $12\text{ cm}\times 9\text{ cm}$ rectangle with a $5\text{ cm}\times 4\text{ cm}$ rectangle removed
b) a square of side $10\text{ m}$ with a triangle of area $18{\text{ m}}^2$ removed

Exercises

Understanding and Fluency

1. Complete each statement:
   a) Area is the amount of ______ inside a shape
   b) Area is measured in ______ units
   c) The area of a rectangle is found by multiplying ______ by ______

2. Convert between metric units of area:
   a) $2{\text{ m}}^2$ to ${\text{cm}}^2$
   b) $70000{\text{ cm}}^2$ to ${\text{m}}^2$
   c) $3\text{ ha}$ to ${\text{m}}^2$
   d) $1.5{\text{ km}}^2$ to $\text{ha}$

3. Convert between metric units of area:
   a) $4500{\text{ m}}^2$ to hectares
   b) $6\text{ ha}$ to ${\text{m}}^2$
   c) $2500000{\text{ m}}^2$ to ${\text{km}}^2$
   d) $0.8{\text{ km}}^2$ to ${\text{m}}^2$

4. Find the area of each shape:
   a) a square with side $5\text{ cm}$
   b) a rectangle with length $12\text{ cm}$ and width $3\text{ cm}$
   c) a parallelogram with base $14\text{ m}$ and height $5\text{ m}$

5. Find the area of each triangle:
   a) base $6\text{ cm}$, height $8\text{ cm}$
   b) base $13\text{ m}$, height $4\text{ m}$
   c) base $20\text{ mm}$, height $9\text{ mm}$

6. Find the area of each shape:
   a) a square with side $9\text{ mm}$
   b) a rectangle with length $7.5\text{ m}$ and width $2\text{ m}$
   c) a parallelogram with base $11\text{ cm}$ and height $6\text{ cm}$

7. Find the area of each composite shape by addition:
   a) a $5\text{ cm}\times 4\text{ cm}$ rectangle joined to a $3\text{ cm}\times 2\text{ cm}$ rectangle
   b) a $9\text{ m}\times 3\text{ m}$ rectangle joined to a $4\text{ m}\times 2\text{ m}$ rectangle
   c) two rectangles of areas $16{\text{ cm}}^2$ and $21{\text{ cm}}^2$

8. Find the area of each composite shape by subtraction:
   a) an $8\text{ cm}\times 7\text{ cm}$ rectangle with a $2\text{ cm}\times 3\text{ cm}$ rectangle removed
   b) a $15\text{ m}\times 10\text{ m}$ rectangle with a $4\text{ m}\times 5\text{ m}$ rectangle removed
   c) a square of side $12\text{ cm}$ with a rectangle of area $30{\text{ cm}}^2$ removed

Reasoning

9. Explain why area is measured in square units instead of units such as cm or m.

10. A student says that the area of a rectangle with length $8\text{ cm}$ and width $5\text{ cm}$ is $13{\text{ cm}}^2$. Explain the mistake.

11. Noah says that the area of a parallelogram is found by multiplying the lengths of its sloping sides. Is he correct? Explain.

12. Explain why the area of a triangle is half the area of a rectangle or parallelogram with the same base and height.

13. A student finds the area of a composite shape by adding all the outside side lengths. Describe the error.

Problem-solving

14. A rectangular garden is $14\text{ m}$ long and $9\text{ m}$ wide. Find its area.

15. A triangular sail has base $6\text{ m}$ and height $4\text{ m}$. Find its area.

16. A parallelogram has base $13\text{ cm}$ and perpendicular height $7\text{ cm}$. Find its area.

17. A school field has area $2\text{ ha}$. Write this area in square metres.

18. A floor plan is made from a $10\text{ m}\times 6\text{ m}$ rectangle and an attached $4\text{ m}\times 3\text{ m}$ rectangle. Find the total area.

19. A $16\text{ m}\times 12\text{ m}$ rectangular block has a $5\text{ m}\times 4\text{ m}$ shed removed from one corner. Find the remaining area.

Potential Misunderstandings

• Students may confuse area with perimeter
• Students may use one-dimensional units for area instead of square units
• Students may confuse length conversions with area conversions
• Students may think $1{\text{ m}}^2=100{\text{ cm}}^2$ instead of $10000{\text{ cm}}^2$
• Students may forget that $1\text{ ha}=10000{\text{ m}}^2$
• Students may use the sloping side of a parallelogram instead of the perpendicular height
• Students may forget to divide by $2$ when finding the area of a triangle
• Students may add side lengths instead of multiplying dimensions when finding area
• Students may not split a composite shape into sensible basic shapes
• Students may add areas when subtraction is needed, or subtract areas when addition is needed