Recover side length from number of cut pieces

3.OA.A.33.MD.D.8 · adapt · grade 3

Archetype: Tile and Cut Figures with Congruent Pieces · step in a 5-type progression

A rectangular sheet of paper is $15\ \text{cm}$ wide. It is cut, with no paper left over, into squares that are each $3\ \text{cm}$ on a side, giving $10$ squares in all. What is the perimeter, in $\text{cm}$ , of the rectangle before it was cut?

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Understand

A rectangular sheet of paper is 15 cm wide. It is cut with no leftover into 3 cm by 3 cm squares, making 10 squares in all. I need the perimeter of the original rectangle.

Givens

The rectangle is 15 cm wide (its top edge is 15 cm).
Each cut square is 3 cm by 3 cm.
Cutting the whole rectangle gives exactly 10 squares with no paper left over.
The figure shows the 15 cm width on top and one 3 cm by 3 cm square in the top-left corner.

Unknowns

The perimeter of the rectangle before it was cut.

Constraints

The squares tile the rectangle exactly (no gaps, no overlap, no leftover).
Both side lengths of the rectangle must be whole multiples of 3 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Break it into parts: first find how many 3 cm squares fit across the 15 cm width (one row), then use the total of 10 squares to find how many rows there are, which gives the other side length. The diagram of squares tiling the rectangle makes these row-and-column counts visible, and the perimeter is then a routine calculation.

Execute

#7 Identify Subproblems 3.OA.A.3

Each square is 3 cm wide and the rectangle is 15 cm wide, so divide to find how many squares fit in one row across the top.

15 \div 3 = 5

How many equal 3 cm pieces fit in 15 cm is a division question.

#7 Identify Subproblems 3.OA.A.3

There are 10 squares in all and 5 squares per row, so divide to find how many rows of squares there are.

10 \div 5 = 2

Total squares split into equal rows gives the number of rows.

#1 Draw a Diagram 3.OA.A.3

Each row is 3 cm tall and there are 2 rows, so multiply to get the rectangle's height (its other side).

2 \times 3 = 6

Stacking 2 squares each 3 cm tall gives a height of 6 cm.

#7 Identify Subproblems 3.MD.D.8

The rectangle is 15 cm by 6 cm. Add the four sides, or double the sum of width and height.

2 \times (15 + 6) = 2 \times 21 = 42

A rectangle's perimeter is twice the length plus twice the width.

Answer: 42 cm

Review

The rectangle is 15 cm by 6 cm. Its area is 15 times 6 equals 90 square cm, and 10 squares of area 3 times 3 equals 9 each also total 90 square cm, so the pieces fit exactly. The perimeter 15 + 6 + 15 + 6 = 42 cm is a sensible length for such a sheet.

Guess and check (tool 6): the height must be a multiple of 3 that makes 5 squares per row times the rows equal 10; only 2 rows (6 cm) works, giving the same 42 cm perimeter.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding squares per row, the number of rows, and the rectangle's height.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of the 15 cm by 6 cm rectangle.

💡 Find the rows and columns of squares, then add up the sides: Grade 3 division and perimeter!