Problem
Perimeter of joined and cut rectangles
A rectangular sheet is 20 inches wide and 12 inches tall. It is cut along one horizontal and three vertical dashed lines. That makes 8 small rectangles. Find the total of all eight perimeters.
Reasoning · Grade 5-1 Perimeter and Area
Your answer
Watch out The spacings look necessary, but each cut edge is counted twice, fixing the total.
How to solve
Strategy Draw a Diagram — The picture is doing the work here, so I redraw it and mark what is actually known: four column widths that must total 20 inches and two row heights that must total 12 inches. Adding eight perimeters one piece at a time is impossible without the individual widths, so I split the job into two easier subproblems and re-sort the information — instead of grouping the sides piece by piece, I group all the horizontal sides together and all the vertical sides together. Sorted that way, the unknown spacings collapse into the two totals I already have. A trial with made-up spacings then confirms the answer really does not depend on where the cuts fall.
1STEP 1
Redraw the figure and record what is actually known
All that is known is 20 wide and 12 tall.
(the 4 column widths) = 20 in, (the 2 row heights) = 12 in
A cut cannot create or destroy width — the pieces of a 20-inch side still add back to 20 inches. Drawing it makes that obvious, and it is the only fact about the spacings we ever need.
4.MD.A.3Draw A Diagram2STEP 2
Split the job: all the horizontal sides, then all the vertical sides
Add the horizontal sides and the vertical sides apart.
total = 2×(sum of the 8 widths) + 2×(sum of the 8 heights)
A perimeter is just its sides added up, so the pieces of a perimeter can be re-sorted into any order without changing the total — and sorting them by direction is the order that makes the unknowns disappear.
3.MD.D.8Identify SubproblemsAdding all the horizontal sides first and all the vertical sides afterwards splits one messy total into two tidy ones.
Why?
Every side of every piece is either horizontal or vertical and never both, so the two piles together are all the sides.
🧱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?
Gathering the sides into two piles cannot change what they come to, because reordering a sum never changes its total.
🧱Commutativity of additionPutting a with b or b with a piles up the same total — order does not change a sum.
3STEP 3
Add every horizontal side
The horizontal sides total 80.
(all horizontal sides) = 20 × 2 rows × 2 = 20 × 4 = 80 in
Reading the widths row by row is what makes them add to something known. Nobody has to know where the three vertical cuts are, only that together the widths span the sheet.
4.NBT.B.5Organize Information In More Ways4STEP 4
Add every vertical side
The vertical sides total 96.
(all vertical sides) = 12 × 4 columns × 2 = 12 × 8 = 96 in
The single horizontal cut is the only thing splitting the height, so each column's two heights must add to 12 inches no matter how high or low that cut was drawn.
4.NBT.B.5Organize Information In More Ways5STEP 5
Add the two parts
Adding gives 176.
80 + 96 = 176 inches
Every side of every piece has been counted exactly once — each piece handed over two widths and two heights — so nothing is double counted and nothing is missed.
4.NBT.B.4Identify Subproblems6STEP 6
Cross-check with the cutting idea
Counting the growth per cut also gives 176.
2×(20+12) + 2×(20 + 3 × 12) = 64 + 2 × 56 = 64 + 112 = 176
Cutting never removes edge, it only adds — and it always adds the cut twice, once for each side of the blade. That single idea handles any pattern of straight cuts.
4.MD.A.3Organize Information In More Ways7STEP 7
Test it with made-up spacings
Any made-up spacings still give 176.
64 + 112 = 176 inches
When a problem refuses to tell you a measurement, that is a hint the answer does not need it — and inventing your own numbers is the quickest way to confirm the hint was right.
4.MD.A.3Guess And CheckAnswer
176 inches
80 + 96 = 176
The answer is a length, so inches is the right unit — a square-inch answer would mean area had been computed by mistake. The size makes sense too: the uncut sheet has a perimeter of only 64 inches, and cutting can only add edge, so the total had to come out bigger than 64, and 176 is bigger. The extra is 176 - 64 = 112 inches, which is exactly twice the 56 inches of cutting that was done, just as the cutting rule predicts. And 176 is comfortably less than what 8 separate full-size sheets would give (8 × 64 = 512 inches), which is right, because the pieces are much smaller than the whole sheet. Finally, working the numbers out with two completely different sets of made-up spacings gave 176 both times, so the answer genuinely does not depend on where the dashed lines sit.
Takeaway
Every cut shows up on both pieces it separates, so the perimeters add up to the sheet's own perimeter plus twice the length you cut.
- Redraw the figure and record what is actually known
- Split the job: all the horizontal sides, then all the vertical sides
- Add every horizontal side
- Add every vertical side
- Add the two parts
- Cross-check with the cutting idea
- Test it with made-up spacings