Perimeter as side length times side count

3.MD.D.83.OA.C.7 · adapt · grade 3

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

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 $\blacksquare\ \text{cm}$ .

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Understand

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.

Plan

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

Trace the staircase outline and count how many 16-cm edges it has to get its perimeter; then set the hexagon's 6 equal sides equal to that total and divide.

Execute

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

Following the heavy outline, the left and right vertical sides are each 3 squares tall, and the stair treads and risers across the top and the 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 16-cm edge turns the staircase shape into a simple count.

#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.

#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

Review

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) being larger than the square side (16 cm) makes sense since 6 hexagon sides must match 12 square edges.

Use units reasoning (tool 8): both perimeters are multiples of equal-length edges; 12 square-edges equal 6 hexagon-edges, so one hexagon edge equals 2 square edges, that is 2 times 16 = 32 cm.

Standards · min 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.

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