Problem
Side length of a square from fitted figures
A big square is cut into smaller squares of different sizes with no gaps. Along every shared edge the lengths must match. The shaded square has a 12 cm side. Find the sides of squares A and B.
Reasoning · Grade 3-1 Shapes and Length
Your answer
Watch out Eyeballing a neighbour's size is tempting — the length of the shared edge fixes it exactly.
How to solve
Strategy Draw a Diagram — In a figure of squares packed edge to edge, an unknown side equals the sum of the sides it rests against. Starting from the known 12 cm shaded square, the square right beside it shares a full edge and is also 12 cm. Then any square resting across two of these 12 cm squares must span 12 + 12. Chaining these edge matches outward reaches both A and B without measuring anything.
1STEP 1
Find the square next to the shaded one
Whatever shares its full edge is also 12 cm.
12 + 12 = 24 cm along the top of the two squares
Two squares that share a whole edge must have the same side length, like two tiles that fit exactly, so side by side they make a 24 cm straight edge.
3.MD.D.8Draw A DiagramTwo squares that share a whole edge must have the same side length, so side by side they make a 24 cm straight edge.
Why?
The shared edge is a side of the first square and also a side of the second, and a square's four sides are all the same length.
🧱Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.
Why?
Laid end to end with no gap and no overlap, the two equal sides make one straight edge whose length is the two added together.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Find square A resting on the two 12 cm squares
Side by side, their tops join into a 24 cm run.
12 + 12 = 24 cm
A square sitting neatly on top of two equal 12 cm squares must be exactly as wide as both of them together, namely 24 cm.
2.NBT.B.5Look For A Pattern3STEP 3
Find the two 12 cm squares beside B
A sits on that run, so A = 24 cm.
12 + 12 = 24 cm along the right of the stacked squares
Stacking two equal 12 cm squares makes a straight 24 cm edge on the side where the next square will touch.
3.MD.D.8Identify Subproblems4STEP 4
Find square B against the stacked squares
On the right two 12s stack, so B = 24 cm.
12 + 12 = 24 cm
B has to be exactly as tall as the two 12 cm squares it leans on, so it is 24 cm on every side.
2.NBT.B.5Draw A DiagramAnswer
A = 24 cm, B = 24 cm
Both A and B come out as 24 cm, which is twice the 12 cm shaded square. That matches the picture, where A and B are clearly larger squares than the shaded one and look about the same size as each other. Each answer is built the same simple way (12 + 12 = 24) from the shaded square's known edge, so the two results agree.
Takeaway
A square that rests on two equal squares is just as wide as both of them together, so 12 + 12 = 24 gives both A and B!
- Find the square next to the shaded one
- Find square A resting on the two 12 cm squares
- Find the two 12 cm squares beside B
- Find square B against the stacked squares