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Use the measurement you know
Task reminder
Two rectangles, two clues. Which clue is area, and which is perimeter?
This rectangle has area 48 cm² and width 6 cm. Find its missing length.
Which side completes the area?
The known area counts square centimetres. The labelled side is six centimetres.
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Figure data
This table is included in print while it is open.
| Quantity | Information |
|---|---|
| Area | 48 cm² |
| Width | 6 cm |
| Length | Not given |
A different rectangle has perimeter 30 cm and width 6 cm. Find its length too. The two questions give different measurements.
Show workingHide answer
For the area, length × 6 = 48, so the length is 8 cm.
For the perimeter, 30 ÷ 2 = 15 cm is one length plus one width. Then 15 − 6 = 9 cm is the length.
The units help choose the relationship: area is in cm²; perimeter and side lengths are in cm. Check a proposed side in the measurement that was actually given.
Check your understanding 1
Keep the area relationship
Check your understanding 1
Keep the area relationship
Use area to find a length
A rectangle covers 13.5 m² and is 3 m wide. Find its length. Check by restoring the area.
Show answerHide answer
13.5 ÷ 3 = 4.5 m. Check 4.5 × 3 = 13.5 m². Halving 13.5 would treat an area as though it were a perimeter.
Find a square’s side
One square has area 49 cm². A different square has perimeter 36 cm. Find the side length of each square.
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The first side is 7 cm, because 7 × 7 = 49 cm². Its perimeter would be 4 × 7 = 28 cm.
The second side is 36 ÷ 4 = 9 cm. Its area would be 9 × 9 = 81 cm².
Dividing an area by four does not find a square’s side. Four sides make the perimeter; two equal dimensions make the area.
Split the shape into rectangles
An L-shaped surface comes from a 10 cm by 8 cm rectangle. A 4 cm by 3 cm piece is removed from its top-right corner. Find the two missing side lengths, then its area and perimeter.
Use the four given lengths
The question marks identify BC and EF. The marked lengths are in centimetres; do not use a screen ruler.
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Figure data
This table is included in print while it is open.
| Side | Length |
|---|---|
| AB | 10 cm |
| BC | Not given |
| CD | 4 cm |
| DE | 3 cm |
| EF | Not given |
| FA | 8 cm |
Check your understanding 2
Choose the surface and the boundary separately
Check your understanding 2
Choose the surface and the boundary separately
Use the lengths in the picture. Area answers count square centimetres; the perimeter follows the six outside sides once.
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BC = 8 − 3 = 5 cm, and EF = 10 − 4 = 6 cm.
The area is 80 − 12 = 68 cm². The removed corner no longer covers the surface.
The perimeter is 10 + 5 + 4 + 3 + 6 + 8 = 36 cm. The inward corner belongs to this boundary too.
Trace the outside for perimeter
The dashed join j splits the same L into two rectangles. It lies inside the surface, so it is not an exposed edge.
One surface, two parts
The lower rectangle spans the full ten-centimetre width. The upper-left rectangle meets it along the dashed join j.
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Figure data
This table is included in print while it is open.
| Segment | Information |
|---|---|
| AB | 10 cm, outside edge |
| BC | 5 cm, outside edge |
| CD | 4 cm, outside edge |
| DE | 3 cm, outside edge |
| EF | 6 cm, outside edge |
| FA | 8 cm, outside edge |
| j | Inside join from the left side to D; find its length from the upper rectangle. |
Find the two areas. Then compare adding their perimeters with tracing the outside of the L.
Show workingHide answer
10 × 5 = 50 cm², and 6 × 3 = 18 cm². Their areas add to 68 cm², with no overlap or gap.
Another valid split gives 6 × 8 = 48 cm² on the left and 4 × 5 = 20 cm² at the bottom right. Their sum is also 68 cm².
The two horizontal-split rectangles have perimeters 30 cm and 18 cm. Adding them gives 48 cm, but each includes the 6 cm join. Remove that inside join twice: 48 − 6 − 6 = 36 cm of exposed boundary.
Subtracting the removed piece’s whole perimeter from the large rectangle’s perimeter would also fail. The corner cut removes 4 cm + 3 cm of old edge and exposes 4 cm + 3 cm of new edge. Here the perimeter stays 36 cm even though the area decreases.
Check your understanding 3
Change where the piece is removed
Check your understanding 3
Change where the piece is removed
Trace around the hole too
A 10 cm by 8 cm card has a 4 cm by 3 cm hole cut wholly inside it. The removed piece has moved away from the corner. Find the remaining area and the length of all exposed edges, including the hole.
Trace both boundary loops
A is the original rectangle. The unshaded rectangle B is cut wholly inside it. Both solid outlines are exposed edges.
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Figure data
This table is included in print while it is open.
| Part | Information |
|---|---|
| Outer rectangle A | 10 cm by 8 cm |
| Hole B | 4 cm by 3 cm |
| Position | The hole is wholly inside the outer rectangle. |
Show answerHide answer
The area is still 80 − 12 = 68 cm². The outside edge is 36 cm. The hole adds 4 + 3 + 4 + 3 = 14 cm. All exposed edges total 36 + 14 = 50 cm.
The question asks for both loops. If it asked for the outside edge only, the answer would be 36 cm. The location of a cut can change boundaries without changing the amount of area removed.
Move the cut to the middle of an edge
Start with a 10 cm by 8 cm card. Remove the same 4 cm by 3 cm piece. This time, cut from the middle of the top edge. Find the remaining area and the length of its outside boundary. All side labels are in centimetres.
Follow the new outline
The cut opens onto the top edge. There is one continuous boundary, with eight sides.
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Figure data
This table is included in print while it is open.
| Side | Length (cm) |
|---|---|
| AB | 10 |
| BC | 8 |
| CD | 3 |
| DE | 3 |
| EF | 4 |
| FG | 3 |
| GH | 3 |
| HA | 8 |
Show workingHide answer
The area remains 80 − 12 = 68 cm².
The top loses a 4 cm edge. The notch adds another 4 cm edge across. It also adds two 3 cm edges down the sides. The outside length is 36 − 4 + 4 + 3 + 3 = 42 cm.
| Cut position | Area (cm²) | Outside edge (cm) | All exposed edges (cm) |
|---|---|---|---|
| Top-right corner | 68 | 36 | 36 |
| Middle of top edge | 68 | 42 | 42 |
| Wholly inside | 68 | 36 | 50 |
The same removed area does not guarantee the same boundary length.
Check your understanding 4
Use the named measurement
Check your understanding 4
Use the named measurement
Finish the garden plan
A rectangular garden has perimeter 42 m and length 13 m. Find its width, then the area of ground it covers. Choose each relationship from the quantity being asked for.
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One length plus one width is 42 ÷ 2 = 21 m. The width is 21 − 13 = 8 m.
Check 13 + 8 + 13 + 8 = 42 m. Then the area is 13 × 8 = 104 m². You used the perimeter to find the missing side. Then you found the area.
Use area to find the perimeter
Use the given area to find the perimeter of the rectangle. Enter an exact number in the labelled unit.
Recover the missing side
Rectangle: use the given area. Not drawn to scale.. Closed boundary through the listed points in order, returning to the first point. Equal drawing scales, y increasing down the page. A (0, 0); B (6, 0); C (6, 3); D (0, 3). Side annotations: A to B: ?; B to C: known.
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Figure data
This table is included in print while it is open.
| Part | Information |
|---|---|
| Shape | Rectangle |
| AB | Unknown side |
| BC | Known side |
| Given | Value |
|---|---|
| Area | 72 cm² |
| Known side | 9 cm |
Show the two relationshipsHide answer
The missing side is 72 ÷ 9 = 8 cm. Check 9 × 8 = 72 cm².
Now follow the boundary: 9 + 8 + 9 + 8 = 34 cm. Use the area to find a side. Then find the length around the edge.
Find the missing fence length
Three sides of a rectangle are fenced. The opposite side is open. Find the full perimeter.
Which side is still open?
Three sides are fenced. The dashed side is open. Not drawn to scale.. Open boundary through the listed points in order. Equal drawing scales, y increasing down the page. Boundary point 1 (0, 0); Boundary point 2 (0, 3); Boundary point 3 (6, 3); Boundary point 4 (6, 0); Line open start (0, 0); Line open end (6, 0). Side annotations: Point 1 to Point 2: end; Point 3 to Point 4: end.
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Figure data
This table is included in print while it is open.
| Part | Information |
|---|---|
| Two fenced end sides | Equal lengths |
| Other fenced side | Opposite the open side |
| Open side | Equal to the opposite fenced side |
| Given | Value |
|---|---|
| Fence across three sides | 37 m |
| Each of the two end sides | 7 m |
Show which sides are countedHide answer
The fence includes 7 m + 7 m at the two ends. The fenced long side is 37 − 7 − 7 = 23 m. The opposite open side is also 23 m.
The whole boundary is 37 + 23 = 60 m. Check all four sides: 7 + 23 + 7 + 23 = 60 m. Doubling 37 would count four end sides. The rectangle has only two.
Check a measurement record
One measurement record is wrong. Choose its row and correct its value and unit.
Figure A
Figure A: rectangular card. AB: 5 cm right; BC: 4 cm up; CD: 5 cm left; DA: 4 cm down.
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Figure data
This table is included in print while it is open.
| Side | Length | Direction |
|---|---|---|
| AB | 5 | right |
| BC | 4 | up |
| CD | 5 | left |
| DA | 4 | down |
Figure B
Figure B: notched card. AB: 10 cm right; BC: 7 cm up; CD: 3 cm left; DE: 2 cm down; EF: 4 cm left; FG: 2 cm up; GH: 3 cm left; HA: 7 cm down.
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Figure data
This table is included in print while it is open.
| Side | Length | Direction |
|---|---|---|
| AB | 10 | right |
| BC | 7 | up |
| CD | 3 | left |
| DE | 2 | down |
| EF | 4 | left |
| FG | 2 | up |
| GH | 3 | left |
| HA | 7 | down |
| Figure | Measurement | Written value |
|---|---|---|
A | area | 20 cm² |
B | perimeter | 34 cm |
Show both checksHide answer
A is correct: its area is 5 × 4 = 20 cm².
For B, the outside rectangle would have perimeter 10 + 7 + 10 + 7 = 34 cm. The notch replaces a 4 cm top edge with a 4 cm base and adds two 2 cm walls. Its perimeter is 34 + 2 + 2 = 38 cm.
Correct B to 38 cm. Keep A’s 20 cm²: it records a covered surface.
Task reminder
Will you keep the same area or the same perimeter?
Keep the covered area
Rectangle A has sides 8 cm and 6 cm. Rectangle B covers the same area, but one side is 12 cm. Find B’s missing side.
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Figure data
This table is included in print while it is open.
| Side | Length (cm) |
|---|---|
| AB | 8 |
| BC | 6 |
| CD | 8 |
| DA | 6 |
Rectangles A and B have the same area. Find that shared area and the missing side of B. Enter exact numbers in the labelled units.
| Given measurement | Value |
|---|---|
| Rectangle A: one side | 8 cm |
| Rectangle A: adjacent side | 6 cm |
| Rectangle B: known side | 12 cm |
Show how the same squares fitHide answer
8 × 6 = 48 cm² in A. B needs 12 × ? = 48, so its missing side is 48 ÷ 12 = 4 cm.
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Figure data
This table is included in print while it is open.
| Side | Length (cm) |
|---|---|
| AB | 12 |
| BC | 4 |
| CD | 12 |
| DA | 4 |
The perimeters differ: A has 28 cm around it; B has 32 cm. Equal area keeps the covered surface, without fixing the boundary length.
Plan another equal-area rectangle
The rectangles below cover equal areas. Find the shared area and B’s missing side. Use the measurement shown. This time, you have no grid.
Rectangles A and B have the same area. Find that shared area and the missing side of B. Enter exact numbers in the labelled units.
| Given measurement | Value |
|---|---|
| Rectangle A: one side | 10 cm |
| Rectangle A: adjacent side | 9 cm |
| Rectangle B: known side | 15 cm |
Show the new area checkHide answer
A covers 10 × 9 = 90 cm². B’s missing side is 90 ÷ 15 = 6 cm. Check 15 × 6 = 90 cm².
Keep the length around the edge
Now A has sides 11 cm and 5 cm. B has the same perimeter, with one side 9 cm. Find B’s missing side.
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Figure data
This table is included in print while it is open.
| Side | Length (cm) |
|---|---|
| AB | 11 |
| BC | 5 |
| CD | 11 |
| DA | 5 |
Rectangles A and B have the same perimeter. Find that shared perimeter and the missing side of B. Enter exact numbers in the labelled units.
| Given measurement | Value |
|---|---|
| Rectangle A: one side | 11 cm |
| Rectangle A: adjacent side | 5 cm |
| Rectangle B: known side | 9 cm |
Show the four-side checkHide answer
A’s perimeter is 11 + 5 + 11 + 5 = 32 cm. One length-and-width pair in B uses 32 ÷ 2 = 16 cm. The missing side is 16 − 9 = 7 cm.
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Figure data
This table is included in print while it is open.
| Side | Length (cm) |
|---|---|
| AB | 9 |
| BC | 7 |
| CD | 9 |
| DA | 7 |
Check 9 + 7 + 9 + 7 = 32 cm. Subtracting just one 9 from 32 would leave three sides together.
The areas differ: A covers 55 cm², while B covers 63 cm². Equal perimeter keeps the boundary length, without fixing the covered surface.
Choose what stays the same
These two rectangles have equal perimeters. Find the perimeter and B’s missing side. Trace the boundary yourself.
Rectangles A and B have the same perimeter. Find that shared perimeter and the missing side of B. Enter exact numbers in the labelled units.
| Given measurement | Value |
|---|---|
| Rectangle A: one side | 12 cm |
| Rectangle A: adjacent side | 6 cm |
| Rectangle B: known side | 14 cm |
Show the new boundary checkHide answer
A’s perimeter is 12 + 6 + 12 + 6 = 36 cm. Half is 18 cm. B’s missing side is 18 − 14 = 4 cm. Check 14 + 4 + 14 + 4 = 36 cm.
Using A’s area instead would preserve a different measurement. Always check the side against the condition the question gives.
For a grown-up
Pip’s synthetic voice uses the University of Edinburgh’s Alba speech dataset, via Piper’s Alba voice, under CC BY 4.0. Audio is optional.
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Syllabus and review details
- Primary 4 Mathematics 2024 · 2021-updated-2025-10
- Primary 5 Foundation Mathematics 2025 · 2021-updated-2025-10
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