Practice · Perimeter as side length times side count · Sensim Math

Variant 1 easy

A shape is made by joining $6$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $6\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $6$ squares, each with side length $6\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $6\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 6 squares (each side 6 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 6 squares, each with side 6 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 6 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 3 edges tall, and the stair treads and risers across the top and bottom add up to 6 more single edges. In total the outline is made of 12 edges, each one square side long.

3 + 3 + 6 = 12 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 12 outline edges is 6 cm, so the perimeter is 12 times 6.

12 \times 6 = 72 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 72 cm, so divide by 6 to get one side.

72 \div 6 = 12 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 12 cm

4 · Reviewdoes it hold up?

Check: a 12 cm hexagon side times 6 sides is 72 cm, equal to the staircase's 12 edges of 6 cm (72 cm). The hexagon side (12 cm) compared with the square side (6 cm) makes sense since 6 hexagon sides must match 12 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 12 square-edges of 6 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 72 cm shared over 6 sides, that is 12 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 12 by 6 and dividing 72 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 2 easy

A shape is made by joining $3$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $9\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $3$ squares, each with side length $9\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $9\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 3 squares (each side 9 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 3 squares, each with side 9 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 9 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 2 edges tall, and the stair treads and risers across the top and bottom add up to 4 more single edges. In total the outline is made of 8 edges, each one square side long.

2 + 2 + 4 = 8 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 8 outline edges is 9 cm, so the perimeter is 8 times 9.

8 \times 9 = 72 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 72 cm, so divide by 6 to get one side.

72 \div 6 = 12 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 12 cm

4 · Reviewdoes it hold up?

Check: a 12 cm hexagon side times 6 sides is 72 cm, equal to the staircase's 8 edges of 9 cm (72 cm). The hexagon side (12 cm) compared with the square side (9 cm) makes sense since 6 hexagon sides must match 8 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 8 square-edges of 9 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 72 cm shared over 6 sides, that is 12 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 8 by 9 and dividing 72 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 3 easy

A shape is made by joining $10$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $6\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $10$ squares, each with side length $6\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $6\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 10 squares (each side 6 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 10 squares, each with side 6 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 6 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 4 edges tall, and the stair treads and risers across the top and bottom add up to 8 more single edges. In total the outline is made of 16 edges, each one square side long.

4 + 4 + 8 = 16 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 16 outline edges is 6 cm, so the perimeter is 16 times 6.

16 \times 6 = 96 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 96 cm, so divide by 6 to get one side.

96 \div 6 = 16 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 16 cm

4 · Reviewdoes it hold up?

Check: a 16 cm hexagon side times 6 sides is 96 cm, equal to the staircase's 16 edges of 6 cm (96 cm). The hexagon side (16 cm) compared with the square side (6 cm) makes sense since 6 hexagon sides must match 16 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 16 square-edges of 6 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 96 cm shared over 6 sides, that is 16 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 16 by 6 and dividing 96 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 4 medium

A shape is made by joining $10$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $9\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $10$ squares, each with side length $9\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $9\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 10 squares (each side 9 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 10 squares, each with side 9 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 9 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 4 edges tall, and the stair treads and risers across the top and bottom add up to 8 more single edges. In total the outline is made of 16 edges, each one square side long.

4 + 4 + 8 = 16 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 16 outline edges is 9 cm, so the perimeter is 16 times 9.

16 \times 9 = 144 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 144 cm, so divide by 6 to get one side.

144 \div 6 = 24 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 24 cm

4 · Reviewdoes it hold up?

Check: a 24 cm hexagon side times 6 sides is 144 cm, equal to the staircase's 16 edges of 9 cm (144 cm). The hexagon side (24 cm) compared with the square side (9 cm) makes sense since 6 hexagon sides must match 16 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 16 square-edges of 9 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 144 cm shared over 6 sides, that is 24 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 16 by 9 and dividing 144 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 5 medium

A shape is made by joining $6$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $12\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $6$ squares, each with side length $12\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $12\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 6 squares (each side 12 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 6 squares, each with side 12 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 12 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 3 edges tall, and the stair treads and risers across the top and bottom add up to 6 more single edges. In total the outline is made of 12 edges, each one square side long.

3 + 3 + 6 = 12 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 12 outline edges is 12 cm, so the perimeter is 12 times 12.

12 \times 12 = 144 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 144 cm, so divide by 6 to get one side.

144 \div 6 = 24 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 24 cm

4 · Reviewdoes it hold up?

Check: a 24 cm hexagon side times 6 sides is 144 cm, equal to the staircase's 12 edges of 12 cm (144 cm). The hexagon side (24 cm) compared with the square side (12 cm) makes sense since 6 hexagon sides must match 12 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 12 square-edges of 12 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 144 cm shared over 6 sides, that is 24 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 12 by 12 and dividing 144 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 6 medium

A shape is made by joining $3$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $15\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $3$ squares, each with side length $15\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $15\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 3 squares (each side 15 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 3 squares, each with side 15 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 15 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 2 edges tall, and the stair treads and risers across the top and bottom add up to 4 more single edges. In total the outline is made of 8 edges, each one square side long.

2 + 2 + 4 = 8 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 8 outline edges is 15 cm, so the perimeter is 8 times 15.

8 \times 15 = 120 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 120 cm, so divide by 6 to get one side.

120 \div 6 = 20 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 20 cm

4 · Reviewdoes it hold up?

Check: a 20 cm hexagon side times 6 sides is 120 cm, equal to the staircase's 8 edges of 15 cm (120 cm). The hexagon side (20 cm) compared with the square side (15 cm) makes sense since 6 hexagon sides must match 8 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 8 square-edges of 15 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 120 cm shared over 6 sides, that is 20 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 8 by 15 and dividing 120 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 7 medium

A shape is made by joining $6$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $16\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $6$ squares, each with side length $16\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $16\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 6 squares (each side 16 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 6 squares, each with side 16 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 16 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 3 edges tall, and the stair treads and risers across the top and bottom add up to 6 more single edges. In total the outline is made of 12 edges, each one square side long.

3 + 3 + 6 = 12 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 12 outline edges is 16 cm, so the perimeter is 12 times 16.

12 \times 16 = 192 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 192 cm, so divide by 6 to get one side.

192 \div 6 = 32 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 32 cm

4 · Reviewdoes it hold up?

Check: a 32 cm hexagon side times 6 sides is 192 cm, equal to the staircase's 12 edges of 16 cm (192 cm). The hexagon side (32 cm) compared with the square side (16 cm) makes sense since 6 hexagon sides must match 12 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 12 square-edges of 16 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 192 cm shared over 6 sides, that is 32 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 12 by 16 and dividing 192 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 8 hard

A shape is made by joining $6$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $18\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $6$ squares, each with side length $18\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $18\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 6 squares (each side 18 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 6 squares, each with side 18 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 18 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 3 edges tall, and the stair treads and risers across the top and bottom add up to 6 more single edges. In total the outline is made of 12 edges, each one square side long.

3 + 3 + 6 = 12 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 12 outline edges is 18 cm, so the perimeter is 12 times 18.

12 \times 18 = 216 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 216 cm, so divide by 6 to get one side.

216 \div 6 = 36 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 36 cm

4 · Reviewdoes it hold up?

Check: a 36 cm hexagon side times 6 sides is 216 cm, equal to the staircase's 12 edges of 18 cm (216 cm). The hexagon side (36 cm) compared with the square side (18 cm) makes sense since 6 hexagon sides must match 12 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 12 square-edges of 18 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 216 cm shared over 6 sides, that is 36 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 12 by 18 and dividing 216 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 9 hard

A shape is made by joining $3$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $21\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $3$ squares, each with side length $21\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $21\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 3 squares (each side 21 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 3 squares, each with side 21 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 21 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 2 edges tall, and the stair treads and risers across the top and bottom add up to 4 more single edges. In total the outline is made of 8 edges, each one square side long.

2 + 2 + 4 = 8 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 8 outline edges is 21 cm, so the perimeter is 8 times 21.

8 \times 21 = 168 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 168 cm, so divide by 6 to get one side.

168 \div 6 = 28 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 28 cm

4 · Reviewdoes it hold up?

Check: a 28 cm hexagon side times 6 sides is 168 cm, equal to the staircase's 8 edges of 21 cm (168 cm). The hexagon side (28 cm) compared with the square side (21 cm) makes sense since 6 hexagon sides must match 8 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 8 square-edges of 21 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 168 cm shared over 6 sides, that is 28 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 8 by 21 and dividing 168 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Variant 10 hard

A shape is made by joining $6$ identical squares (no overlaps), and another shape is made by joining $6$ triangles whose sides are all equal (no overlaps). The side length of each square is $24\ \text{cm}$ . When the length of the heavy (bold) outline of the two shapes is the same, find the value of $\blacksquare$ .

(Figure) The left shape is a staircase formed by joining $6$ squares, each with side length $24\ \text{cm}$ ; the heavy outline around the outside is its perimeter, and one square's horizontal and vertical sides are each labeled $24\ \text{cm}$ . The right shape is a regular hexagon formed by joining $6$ equilateral triangles; dashed segments run from the center to each vertex, and the heavy outline (the hexagon's six sides) is its perimeter. One bottom side of the hexagon is labeled $\blacksquare\ \text{cm}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A staircase made of 6 squares (each side 24 cm) and a regular hexagon made of 6 equilateral triangles have heavy outlines of equal length. Knowing the square's side, we find the hexagon's side, marked with the blank.

Givens

The left shape joins 6 squares, each with side 24 cm.
Its heavy outline (perimeter) is the staircase boundary shown.
The right shape is a regular hexagon of 6 equilateral triangles; all six sides are equal.
The two heavy outlines have the same total length.
One hexagon side is labeled with the blank, in cm.

Unknowns

The length of one hexagon side (the blank), in cm.

Constraints

Each staircase edge segment is one square side, 24 cm long.
The hexagon is regular, so all six sides are equal.

2 · Planchoose the strategy

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

Trace the staircase outline and count how many same-length edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

3 · Execute3 carry out the plan

1Count the staircase outline edges

#1 Draw a Diagram 3.MD.D.8

Following the heavy outline, the left and right vertical sides are each 3 edges tall, and the stair treads and risers across the top and bottom add up to 6 more single edges. In total the outline is made of 12 edges, each one square side long.

3 + 3 + 6 = 12 \text{ edges}

Walking around the boundary and counting each edge turns the staircase shape into a simple count.

2Compute the staircase perimeter

#7 Identify Subproblems 3.OA.C.7

Each of the 12 outline edges is 24 cm, so the perimeter is 12 times 24.

12 \times 24 = 288 \text{ cm}

Multiplying the edge count by the edge length gives the whole heavy outline at once.

3Find the hexagon side

#7 Identify Subproblems 3.MD.D.8

The regular hexagon's 6 equal sides have the same total length 288 cm, so divide by 6 to get one side.

288 \div 6 = 48 \text{ cm}

Equal perimeters mean the hexagon's six equal sides share the same total, so each side is one sixth of it.

Answer: 48 cm

4 · Reviewdoes it hold up?

Check: a 48 cm hexagon side times 6 sides is 288 cm, equal to the staircase's 12 edges of 24 cm (288 cm). The hexagon side (48 cm) compared with the square side (24 cm) makes sense since 6 hexagon sides must match 12 square edges.

Another way: Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 12 square-edges of 24 cm equal 6 hexagon-edges, so one hexagon edge is the shared total 288 cm shared over 6 sides, that is 48 cm.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the staircase perimeter and equating it to the hexagon's perimeter.
3.OA.C.7 Fluently multiply and divide within 100 — Multiplying 12 by 24 and dividing 288 by 6.

💡Takeaway. This only needs Grade 3 perimeter sense: count the edges, multiply, then share the total over the hexagon's 6 sides!

Perimeter as side length times side count · 10 practice problems

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