Practice · Find a side from equal perimeters · Sensim Math

Variant 1 answer: 18 cm

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 $24\,\text{cm}$ wide and $12\,\text{cm}$ 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 24 cm wide and 12 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 24 cm wide and 12 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 24 cm and two sides of 12 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 24 cm and two sides of 12 cm. Add all four sides to get the perimeter.

24 + 12 + 24 + 12 = 2 \times (24 + 12) = 72

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 72 cm.

P_B = P_A = 72 \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.

72 \div 4 = 18

Finding the side from 'four equal sides add to 72' is the Grade 3 unknown-factor idea: 4 x ? = 72.

Answer: 18 cm

Review

Square B's side 18 cm sits between A's two side lengths 12 cm and 24 cm, which makes sense because a square balances A's long and short sides. Checking: 4 x 18 = 72 cm equals A's perimeter 2 x (24 + 12) = 72 cm.

Guess and check (tool 6): try side 17 -> 68 cm (too small), side 19 -> 76 cm (too big), side 18 -> 72 cm (just right), confirming 18 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) = perimeter 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!

Variant 2 answer: 10 cm

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) = perimeter 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!

Variant 3 answer: 12 cm

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 $14\,\text{cm}$ wide and $10\,\text{cm}$ 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 14 cm wide and 10 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 14 cm wide and 10 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 14 cm and two sides of 10 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 14 cm and two sides of 10 cm. Add all four sides to get the perimeter.

14 + 10 + 14 + 10 = 2 \times (14 + 10) = 48

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 48 cm.

P_B = P_A = 48 \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.

48 \div 4 = 12

Finding the side from 'four equal sides add to 48' is the Grade 3 unknown-factor idea: 4 x ? = 48.

Answer: 12 cm

Review

Square B's side 12 cm sits between A's two side lengths 10 cm and 14 cm, which makes sense because a square balances A's long and short sides. Checking: 4 x 12 = 48 cm equals A's perimeter 2 x (14 + 10) = 48 cm.

Guess and check (tool 6): try side 11 -> 44 cm (too small), side 13 -> 52 cm (too big), side 12 -> 48 cm (just right), confirming 12 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) = perimeter 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!

Variant 4 answer: 8 cm

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 $9\,\text{cm}$ wide and $7\,\text{cm}$ 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 9 cm wide and 7 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 9 cm wide and 7 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 9 cm and two sides of 7 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 9 cm and two sides of 7 cm. Add all four sides to get the perimeter.

9 + 7 + 9 + 7 = 2 \times (9 + 7) = 32

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 32 cm.

P_B = P_A = 32 \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.

32 \div 4 = 8

Finding the side from 'four equal sides add to 32' is the Grade 3 unknown-factor idea: 4 x ? = 32.

Answer: 8 cm

Review

Square B's side 8 cm sits between A's two side lengths 7 cm and 9 cm, which makes sense because a square balances A's long and short sides. Checking: 4 x 8 = 32 cm equals A's perimeter 2 x (9 + 7) = 32 cm.

Guess and check (tool 6): try side 7 -> 28 cm (too small), side 9 -> 36 cm (too big), side 8 -> 32 cm (just right), confirming 8 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) = perimeter 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!

Variant 5 answer: 10 cm

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 $16\,\text{cm}$ wide and $4\,\text{cm}$ 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 16 cm wide and 4 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 16 cm wide and 4 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 16 cm and two sides of 4 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 16 cm and two sides of 4 cm. Add all four sides to get the perimeter.

16 + 4 + 16 + 4 = 2 \times (16 + 4) = 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 4 cm and 16 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 (16 + 4) = 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) = perimeter 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!

Variant 6 answer: 12 cm

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 $18\,\text{cm}$ wide and $6\,\text{cm}$ 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 18 cm wide and 6 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 18 cm wide and 6 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 18 cm and two sides of 6 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 18 cm and two sides of 6 cm. Add all four sides to get the perimeter.

18 + 6 + 18 + 6 = 2 \times (18 + 6) = 48

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 48 cm.

P_B = P_A = 48 \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.

48 \div 4 = 12

Finding the side from 'four equal sides add to 48' is the Grade 3 unknown-factor idea: 4 x ? = 48.

Answer: 12 cm

Review

Square B's side 12 cm sits between A's two side lengths 6 cm and 18 cm, which makes sense because a square balances A's long and short sides. Checking: 4 x 12 = 48 cm equals A's perimeter 2 x (18 + 6) = 48 cm.

Guess and check (tool 6): try side 11 -> 44 cm (too small), side 13 -> 52 cm (too big), side 12 -> 48 cm (just right), confirming 12 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) = perimeter 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!

Variant 7 answer: 10 cm

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 $15\,\text{cm}$ wide and $5\,\text{cm}$ 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 15 cm wide and 5 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 15 cm wide and 5 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 15 cm and two sides of 5 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 15 cm and two sides of 5 cm. Add all four sides to get the perimeter.

15 + 5 + 15 + 5 = 2 \times (15 + 5) = 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 5 cm and 15 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 (15 + 5) = 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) = perimeter 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!

Variant 8 answer: 14 cm

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 $20\,\text{cm}$ 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 20 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 20 cm wide and 8 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 20 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 20 cm and two sides of 8 cm. Add all four sides to get the perimeter.

20 + 8 + 20 + 8 = 2 \times (20 + 8) = 56

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 56 cm.

P_B = P_A = 56 \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.

56 \div 4 = 14

Finding the side from 'four equal sides add to 56' is the Grade 3 unknown-factor idea: 4 x ? = 56.

Answer: 14 cm

Review

Square B's side 14 cm sits between A's two side lengths 8 cm and 20 cm, which makes sense because a square balances A's long and short sides. Checking: 4 x 14 = 56 cm equals A's perimeter 2 x (20 + 8) = 56 cm.

Guess and check (tool 6): try side 13 -> 52 cm (too small), side 15 -> 60 cm (too big), side 14 -> 56 cm (just right), confirming 14 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) = perimeter 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!

Variant 9 answer: 8 cm

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 $10\,\text{cm}$ wide and $6\,\text{cm}$ 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 10 cm wide and 6 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 10 cm wide and 6 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 10 cm and two sides of 6 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 10 cm and two sides of 6 cm. Add all four sides to get the perimeter.

10 + 6 + 10 + 6 = 2 \times (10 + 6) = 32

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 32 cm.

P_B = P_A = 32 \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.

32 \div 4 = 8

Finding the side from 'four equal sides add to 32' is the Grade 3 unknown-factor idea: 4 x ? = 32.

Answer: 8 cm

Review

Square B's side 8 cm sits between A's two side lengths 6 cm and 10 cm, which makes sense because a square balances A's long and short sides. Checking: 4 x 8 = 32 cm equals A's perimeter 2 x (10 + 6) = 32 cm.

Guess and check (tool 6): try side 7 -> 28 cm (too small), side 9 -> 36 cm (too big), side 8 -> 32 cm (just right), confirming 8 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) = perimeter 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!

Variant 10 answer: 10 cm

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 $11\,\text{cm}$ wide and $9\,\text{cm}$ 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 11 cm wide and 9 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 11 cm wide and 9 cm tall.
A rectangle's opposite sides are equal, so A has two sides of 11 cm and two sides of 9 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 11 cm and two sides of 9 cm. Add all four sides to get the perimeter.

11 + 9 + 11 + 9 = 2 \times (11 + 9) = 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 9 cm and 11 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 (11 + 9) = 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) = perimeter 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!

Find a side from equal perimeters · 10 practice problems

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