Base perimeter from multiple views

3.MD.D.8K.G.A.34.MD.A.3 · adapt · grade 4

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

Using $11$ of the unit cubes shown on the left, a solid is built as shown on the right. What is the perimeter of the part that touches the floor, in $\text{cm}$ ?

Figure description: On the left is a single cube-shaped block with an edge length of $1\,\text{cm}$ . On the right is a solid built by joining $11$ of these cubes together with no gaps. The cubes touching the floor are arranged in a staircase-like footprint, and a few of the squares have one more cube stacked on top of them. The part that touches the floor is a flat figure made of joined squares, each with a side length of $1\,\text{cm}$ .

Show solution

Understand

A solid is built from 11 unit cubes (each 1 cm on a side). I read the picture to find which squares touch the floor, then measure the perimeter of that flat floor-footprint figure.

Givens

Each unit cube has an edge length of 1 cm.
The solid uses 11 unit cubes joined with no gaps.
The floor-touching cubes form a staircase-like footprint; a few squares have one extra cube stacked on top.
From the figure, the floor footprint is an L-shape: a row of 6 unit squares with a row of 3 unit squares attached beneath its left end (9 floor squares total, 2 cubes stacked above).

Unknowns

The perimeter, in cm, of the flat figure where the solid touches the floor.

Constraints

Only the cubes resting on the floor count toward the footprint; stacked cubes do not add to it.
Each footprint square has side 1 cm.

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

First I mentally view the 3D solid from below to see exactly which squares press on the floor (spatial visualization). I redraw that footprint flat on paper, then break the perimeter into its straight outer edges and add their 1-cm lengths. Stacked cubes are ignored because they sit above the floor.

Execute

#17 Visualize Spatial Relationships K.G.A.3

Looking at the solid from below, 9 of the 11 cubes rest on the floor and 2 are stacked on top of floor cubes. The 9 floor squares form an L-shape: a top row of 6 unit squares and, under its left 3 squares, a bottom row of 3 unit squares.

9 + 2 = 11

Telling apart the flat floor shape (2D) from the stacked cubes (3D) is the kindergarten idea of two- vs three-dimensional shapes.

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

Drawing the L-shape on grid paper: it fits inside a 6 cm by 2 cm rectangle, with a 3 cm by 1 cm notch missing from the bottom-right. Each side of every unit square is 1 cm.

6 \times 2 = 12 \text{ squares fit the box, } 12 - 3 = 9 \text{ are filled}

Sketching the flat outline turns a 3D puzzle into a simple polygon-perimeter problem at the Grade 3 level.

#7 Identify Subproblems 4.MD.A.3

Walk the outline of the L-shape, counting 1-cm edges: across the top is 6, down the right is 1, left along the step is 3, down 1, left along the bottom is 3, and up the left side is 2. Adding these gives the perimeter.

6 + 1 + 3 + 1 + 3 + 2 = 16

For an L made of two rectangles, the inner step lengths slide together so the perimeter equals that of the 6 by 2 bounding rectangle: 2 x (6 + 2) = 16.

Answer: 16 cm

Review

The footprint fits in a 6 cm by 2 cm box whose perimeter is 2 x (6 + 2) = 16 cm; an L-shape carved from that box keeps the same perimeter because the notch edges fold inward without adding length. The value 16 cm is a whole number of centimeters, which fits a figure made of 1 cm unit squares.

Count unit-edges directly (tool 2): list each of the 9 squares and mark every side that is not shared with a neighbor; counting those exposed 1-cm sides gives 16, the same answer.

Standards · min grade 4

K.G.A.3 Identify shapes as two-dimensional or three-dimensional — Separating the flat floor footprint (2D) from the stacked cubes (3D).
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting up and finding the perimeter of the L-shaped footprint polygon.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Using the bounding-rectangle perimeter 2 x (6 + 2) to compute and check 16 cm.

💡 Look at the solid from underneath, draw the flat shape it sits on, then add the outside edges - an L-shape's perimeter matches its bounding box!