The idea in brief. When the pieces are identical, a length you cannot measure can be written two ways, and the two ways must agree; so name one piece's breadth 1 unit, count the figure in units, and let the two readings fix what a unit is.
Lesson video, with the working
Identical pieces: find the missing measure
4 min 57 s
The film teaches 2 questions in turn. Each question shows first; its working is written as the film explains it.
Five identical rectangles are placed along the straight line AB, three of them above it and two below. Above the line the three rectangles leave gaps of 4 cm and 7 cm between them. Below the line the two rectangles leave gaps of 9 cm, 13 cm and 6 cm. Find the length of each rectangle.
The two gaps in the top row = 4 cm + 7 cm = 11 cm
Along the top row, AB = 3 lengths + 11 cm
The three gaps in the bottom row = 9 cm + 13 cm + 6 cm = 28 cm
Along the bottom row, AB = 2 lengths + 28 cm
AB is the same line both ways.
3 lengths + 11 cm = 2 lengths + 28 cm
1 length = 28 cm − 11 cm = 17 cm
Ans: 17 cm
Example 1
Five identical rectangles are fitted together to make one larger rectangle: one lies across the top, one lies across the bottom, and three stand upright side by side between them. The perimeter of each small rectangle is 72 cm. Find the area of the figure.
The width of the figure is 1 length, and also 3 breadths.
1 length = 3 breadths
The perimeter of one rectangle = 2 lengths + 2 breadths = 8 breadths
8 breadths = 72 cm
1 breadth = 72 cm ÷ 8 = 9 cm
1 length = 3 × 9 cm = 27 cm
Area of one rectangle = 27 cm × 9 cm = 243 cm²
Area of the figure = 5 × 243 cm² = 1215 cm²
Ans: 1215 cm²
How to solve it
Read AB along both rows. The top has 3 lengths plus 11 centimetres; the bottom has 2 lengths plus 28 centimetres.
The same line has the same length. One rectangle length is 28 minus 11, which is 17 centimetres.
In the second figure, 1 length equals 3 breadths. The perimeter of one rectangle is therefore 8 breadths.
Find a breadth: 72 divided by 8 is 9 centimetres. A length is 3 times 9, which is 27 centimetres.
Find one rectangle's area: 27 times 9 is 243 square centimetres.
The figure has five equal areas. 5 times 243 is 1215 square centimetres.
Read the same length two ways. Use identical pieces to find the missing measure.
Practice questions with worked solutions
Question 1 is free to try. The other 6 open in the free trial, each with its full working.
Question 1 of 7 · Basic · free to tryThe figure is made up of 6 identical rectangles arranged in 2 rows of 3…The figure is made up of 6…
The figure is made up of 6 identical rectangles arranged in 2 rows of 3, each lying on its long side. The length of each rectangle is twice its breadth and the perimeter of each rectangle is 36 cm. Find the perimeter of the figure.
The perimeter of one rectangle is made up of 6 units.
6 units = 36 cm
1 unit = 36 cm ÷ 6 = 6 cm
Length of each rectangle = 2 × 6 cm = 12 cm
Length of the figure = 3 × 12 cm = 36 cm
Breadth of the figure = 2 × 6 cm = 12 cm
Perimeter = 36 cm + 12 cm + 36 cm + 12 cm = 96 cm
The perimeter of the figure is 96 cm.
Answer: 96 cm
For parents
How can I help my child solve identical rectangles questions?
When a figure is built from identical rectangles or squares, one piece's dimensions apply to every other copy, including copies that have been turned.
What your child learns here
When a figure is built from identical rectangles or squares, one piece's dimensions apply to every other copy, including copies that have been turned. Your child uses this repeated information to find missing lengths before calculating. These questions apply the Primary 4 composite-figure outcome, with the extra task of recognising how one rectangle's length can line up with several breadths elsewhere in the arrangement.
Transfer a known length or breadth between pieces only when the question states that those pieces are identical.
Read a total distance as a sum of the component sides that lie along it, taking account of rotated rectangles.
Use equal total lengths to recover an unknown dimension, then find the area of one piece from its length and breadth.
Multiply one piece's area by the number of non-overlapping identical pieces for total area, or count exposed sides for perimeter.
Does the answer make sense?
If five identical rectangles form a figure without overlapping, its total area must be five times the area of one rectangle. Your child can divide the proposed total by five and compare it with length × breadth for a single piece, using the dimensions already found.
How to practise these questions
Basic questions connect a repeated rectangle's dimensions with the full perimeter or find the perimeter of a cross made from five squares. Intermediate questions use a row's perimeter to find its area or recover square sizes from a sum and difference. Challenge work includes a square assembled from rectangles, two rows of differently sized squares with matching total lengths, and a chain of dimensions across squares and turned rectangles. Begin each attempt by identifying which pieces are identical and which complete distances are equal. Your child can then derive the individual dimensions before calculating the requested area or perimeter. When checking, compare each recovered length with every relation that uses it, since a length consistent with only one part of the arrangement may still be unsuitable.
Questions parents ask
What does identical tell my child about the pieces? It means the pieces have the same shape and size, so corresponding lengths match. Turning a rectangle changes which direction its length points, but does not exchange its actual length and breadth values. Ask your child to identify a long side and a short side on each copy before using a measurement from one piece on another.
How can one side be described in two different ways? An aligned distance might run along one long side in one part of the figure and along several short sides in another. Both descriptions measure the same endpoints, so their totals must agree. Your child should establish that shared distance from the arrangement before making a calculation; nearby edges do not necessarily span the same distance.
Why is the perimeter not the number of pieces times one perimeter? Joining the pieces hides some edges inside the combined figure. Multiplying one perimeter counts those joins even though they are no longer on the outside boundary. Your child can still use repeated side lengths to simplify the work, but must count how many long and short sides are exposed in the particular arrangement shown.
