Problem
Find a side from equal perimeters
Rectangle A is 12 cm wide and 8 cm tall. Square B has the same perimeter (total of its four side lengths) as rectangle A. I must find the length of one side of square B.
Measurement & dataOperations
Your answer
cm
Watch out It's easy to forget a rectangle has two of each side - treating 12 + 8 as the whole perimeter (instead of doubling it) gives the wrong side length for square B.
How to solve
Strategy Identify Subproblems — Break the task into two easy steps: first compute the perimeter of rectangle A, then split that perimeter into 4 equal square sides. A quick sketch of both shapes makes clear which lengths repeat (two widths, two heights for A; four equal sides for B).
1STEP 1
Find the perimeter of rectangle A
Rectangle A has two sides of 12 cm and two sides of 8 cm. Add all four sides to get the perimeter.
12 + 8 + 12 + 8 = 2 × (12 + 8) = 40
Adding the four sides of a rectangle is the basic Grade 3 idea of perimeter.
3.MD.D.8Identify Subproblems2STEP 2
Match square B's perimeter
Square B has the same perimeter as A, so square B's four sides also add to 40 cm.
P_B = P_A = 40 cm
Equal perimeters just means the two outlines are the same total length - easy to see by sketching both shapes.
4.MD.A.3Draw A Diagram3STEP 3
Split into four equal sides
A square's four sides are equal, so divide its perimeter by 4 to find one side.
40 ÷ 4 = 10
Finding the side from 'four equal sides add to 40' is the Grade 3 unknown-factor idea: 4 x ? = 40.
3.OA.A.4Identify SubproblemsThe length of one side of square B is its perimeter divided by four.
Why?
The square's perimeter is its four equal sides joined end to end, so the perimeter is four times the length of one side.
Why?
The perimeter is the whole distance around the square, and the four sides are exactly the parts that make up that whole with no gaps or overlaps.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
All four sides are the same length, so adding the four sides is the same as four equal groups of one side, which is four times that side.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Because four times one side makes the whole perimeter, splitting the perimeter back into four equal parts returns one side, since division reverses multiplication.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Answer
10 cm
40 ÷ 4 = 10
Square B's side 10 cm sits between A's two side lengths 8 cm and 12 cm, which makes sense because a square balances A's long and short sides. Checking: 4 x 10 = 40 cm equals A's perimeter 2 x (12 + 8) = 40 cm.
Takeaway
Add the rectangle's four sides, then share that total equally among the square's four sides - just Grade 3 perimeter and division!
- Find the perimeter of rectangle A
- Match square B's perimeter
- Split into four equal sides
Where next?
Level upsuggested
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