Boundary and area problems | Year 5

01 / LEARN

Understand the idea.

Boundary or inside?

Perimeter measures the boundary; area measures the surface inside. Decide where the material goes: edging follows a boundary, while tiles cover a surface. Use this decision to choose a calculation.

A rectangle has two lengths and two widths, so add all four sides for perimeter. Multiply length by width for area. Perimeter has length units, such as metres; area has square units, such as $m^{2}$ .

Finish the practical answer

Complete the practical question after finding the shape measure. If a gate needs no fencing, subtract its width from the full perimeter. If a surface has a hole, subtract the missing area from the full area.

An equal perimeter does not force equal area. Calculate both measures when comparing shapes. Moving the same amount of edging into a longer, thinner rectangle can change how much surface it encloses.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Fence with a gate

An 8 m by 5 m rectangular garden has a 2 m gate opening that needs no fence. Find the fence length.

Worked solution

1
Add the garden boundary lengths.
$8 + 5 + 8 + 5 = 26 m$
Opposite sides of the rectangular garden have equal lengths.
2
Remove the unfenced gate opening.
$26 - 2 = 24 m$
The gate occupies 2 m of the boundary but needs no fence.
3
Check that the boundary is accounted for.
$24 + 2 = 26 m$
Fence length plus gate width should equal the full perimeter.

Answer

24 m

EXAMPLE 02

Same perimeter, different area

Compare a square with side 4 cm and a 6 cm by 2 cm rectangle.

Worked solution

1
Find the square perimeter and area.
$\begin{aligned} P & = 4 + 4 + 4 + 4 = 16 cm \\ A & = 4 \times 4 = 16 {cm}^{2} \end{aligned}$
The square has four equal sides and a 4-by-4 square-unit covering.
2
Find the rectangle perimeter and area.
$\begin{aligned} P & = 6 + 2 + 6 + 2 = 16 cm \\ A & = 6 \times 2 = 12 {cm}^{2} \end{aligned}$
The rectangle has two 6 cm sides and two 2 cm sides.
3
Compare the two measures separately.
$16 - 12 = 4 {cm}^{2}$
Both perimeters are 16 cm, while the square encloses 4 square centimetres more.

Answer

Both perimeters are 16 cm; the square has 4 cm² more area.

03 / YOUR TURN

Try it yourself.

Check your understanding

A photo is 9 cm by 6 cm. How much edging goes once around it?

Show answer & reasoning

30 cm

Edging goes around the photo, so add both lengths and both widths: $9 + 6 + 9 + 6 = 30 cm$ . Multiplying would find its area instead.

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Find the perimeter of a 7 cm by 3 cm rectangle.
2 marks
Find the perimeter of a square with side 6 m.
2 marks
A rectangle has perimeter 24 cm and width 4 cm. Find its length.
3 marks
A 10 m by 4 m garden has a 3 m opening. Find the fencing length.
2 marks
Choose area or perimeter for each task.
2 marks
1. (a)
  Carpet to cover a room.
  1 mark
2. (b)
  Ribbon around a card.
  1 mark
Compare a 5 cm by 5 cm square and a 7 cm by 3 cm rectangle. Check their perimeters and compare their areas.
4 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the perimeter of a 7 cm by 3 cm rectangle.
2 marks
$7 + 3 + 7 + 3 = 20$ cm.
20 cm
Find the perimeter of a square with side 6 m.
2 marks
$4 \times 6 = 24$ m.
24 m
A rectangle has perimeter 24 cm and width 4 cm. Find its length.
3 marks
Two widths: $2 \times 4 = 8$ cm.
Two lengths: $24 - 8 = 16$ cm.
One length: $16 \div 2 = 8$ cm.
8 cm
A 10 m by 4 m garden has a 3 m opening. Find the fencing length.
2 marks
Perimeter: $10 + 4 + 10 + 4 = 28$ m.
$28 - 3 = 25$ m.
25 m
Choose area or perimeter for each task.
2 marks
Carpet covers an inside surface.
Ribbon follows an outside boundary.
1. (a)
  Carpet to cover a room.
  1 mark
  Carpet covers the inside surface.
  Area
2. (b)
  Ribbon around a card.
  1 mark
  Ribbon follows the outside boundary.
  Perimeter
Carpet needs area; ribbon around a card needs perimeter.
Compare a 5 cm by 5 cm square and a 7 cm by 3 cm rectangle. Check their perimeters and compare their areas.
4 marks
Square: perimeter $4 \times 5 = 20$ cm; area $5 \times 5 = 25$ ${cm}^{2}$ .
Rectangle: perimeter $7 + 3 + 7 + 3 = 20$ cm; area $7 \times 3 = 21$ ${cm}^{2}$ .
Both perimeters are 20 cm. The areas are 25 ${cm}^{2}$ and 21 ${cm}^{2}$ , so the square has 4 ${cm}^{2}$ more area.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics 2026

Units/precision; regular/irregular regions; boundary and area problems.

Official curriculum source ↗

In this lesson

Understand the idea.

Boundary or inside?

Finish the practical answer

See the method in action.

Fence with a gate

Add the garden boundary lengths.

Remove the unfenced gate opening.

Check that the boundary is accounted for.

Same perimeter, different area

Find the square perimeter and area.

Find the rectangle perimeter and area.

Compare the two measures separately.

Try it yourself.

Check your understanding

Keep practising