Find a side from equal perimeters

3.MD.D.84.MD.A.33.OA.A.4 · adapt · grade 4

Archetype: Perimeter by Tracing Every Side · step in a 11-type progression

The perimeter of rectangle $A$ (the sum of the lengths of its four sides) equals the perimeter of square $B$ . How many $\text{cm}$ long is one side of square $B$ ?

Rectangle $A$ is $12\,\text{cm}$ wide and $8\,\text{cm}$ tall. Square $B$ has four sides of equal length, and the sum of its four side lengths equals the sum of the four side lengths of rectangle $A$ .

Show solution

Understand

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.

Givens

Rectangle A is 12 cm wide and 8 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 12 cm and two sides of 8 cm.
Square B has four equal sides.
The perimeter of square B equals the perimeter of rectangle A.

Unknowns

The length of one side of square B, in cm.

Constraints

All four sides of a square are the same length.
Perimeter means the sum of all four side lengths.

Plan

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

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).

Execute

#7 Identify Subproblems 3.MD.D.8

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 \times (12 + 8) = 40

Adding the four sides of a rectangle is the basic Grade 3 idea of perimeter.

#1 Draw a Diagram 4.MD.A.3

Square B has the same perimeter as A, so square B's four sides also add to 40 cm.

P_B = P_A = 40 \text{ cm}

Equal perimeters just means the two outlines are the same total length - easy to see by sketching both shapes.

#7 Identify Subproblems 3.OA.A.4

A square's four sides are equal, so divide its perimeter by 4 to find one side.

40 \div 4 = 10

Finding the side from 'four equal sides add to 40' is the Grade 3 unknown-factor idea: 4 x ? = 40.

Answer: 10 cm

Review

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.

Guess and check (tool 6): try side 9 -> 36 cm (too small), side 11 -> 44 cm (too big), side 10 -> 40 cm (just right), confirming 10 cm.

Standards · min grade 4

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Computing the perimeter of rectangle A by adding its four sides.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Equating the two perimeters and reasoning with the perimeter relationship.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Solving 4 x (side) = 40 to find one side of the square.

💡 Add the rectangle's four sides, then share that total equally among the square's four sides - just Grade 3 perimeter and division!