▦Math Education · Grade 5
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Chapter 6 · Solid-Figure Problems

Lesson 6.1 — Count Unit Cubes, Layers, and Exposed Faces

第6讲 · 立体图形问题

Build a solid in your mind—or on the screen—then count exactly what the question asks: cubes for volume, exposed square faces for surface area.

See what is hidden. Count what is exposed.
Grade 5 enrichment40–55 minutesNo textbook requiredInteractive solid models
Mission 1

Speak the language of a cube

A unit cube has edge length 1 unit. Its volume is 1 cubic unit, and each of its six square faces has area 1 square unit.

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One unit cubeDiagram not to scale
face = square regionedge = line segment

A vertex is a corner where edges meet. A cube has 6 faces, 12 edges, and 8 vertices. A drawing shows only some of them. The hidden ones still exist.

Two different questions

Volume asks how much space the solid occupies. For a unit-cube model, count the cubes.

Volume in cubic units = number of unit cubes

Surface area asks how much outside covering is needed. Count faces not touching another cube, including the underside even when the model rests on a table.

Surface area in square units = number of exposed unit faces
Do not mix them. A hidden cube still adds volume, but a face touching another cube does not add surface area.

Quick check

Mission 2

Count the staircase by horizontal layers

Our introductory staircase has four square layers, stacked without gaps and aligned at the same corner. Each cube has edge length 1 cm. A layer view turns a complicated perspective drawing into four simple counts.

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Four-layer unit-cube staircaseShowing layers 1–4
4 layers

Move the slider toward 1 to peel away upper layers. Answer the questions for the complete four-layer staircase.

Layer 1

A 4×4 square:

4×4 = 16 cubes

Layer 2

A 3×3 square:

3×3 = 9 cubes

Layer 3

A 2×2 square:

2×2 = 4 cubes

Layer 4

A 1×1 square:

1 cube
Layer total: 16+9+4+1=30. Hidden cubes are included because every cube belongs to exactly one horizontal layer.

Build the total

Mission 3

A top-view map can reveal hidden cubes

Each number in a top-view square tells how many cubes are stacked in that vertical column, starting at ground level with no gaps. Add the column heights—not merely the visible top squares.

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Top-view height map

2
1
0
1
3
1
0
1
2

Back row at the top; front row at the bottom. A 0 means there is no cube in that column.

2+1+0+1+3+1+0+1+2 = 11 cubes
Build the height map11 cubes
All 3 layers

The numbers on the roofs show the column heights currently displayed. Answer the questions using the complete three-layer model and the fixed map. Lower the slider to peel away upper layers: the bottom layer has 7 cubes, the second has 3, and the top has 1. The back row of the map is farthest from you; the front row is nearest.

Read the map

Mission 4

Every face is either exposed or hidden by a neighbour

Start with six faces per cube. Every face-to-face contact hides two faces: one from each cube.

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Three cubes in a rowSurface area = 14

Contact-pair method

Three separate cubes would have:

3×6 = 18 faces

There are two contacts. Each contact hides two faces:

18 − 2×2 = 14 exposed faces

For cubes on a unit grid, count only contacts along an entire square face; touching at an edge or corner hides no square face. Then:

Surface area = 6×cubes − 2×contacts

Count contacts before faces

Mission 5

Count the outside of the staircase by direction

The same 30-cube staircase has 72 square centimetres of surface area. Count its top, bottom, and four side directions so no face is missed or counted twice.

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Directional surface countLesson staircase

The drawing shows the top and two visible sides. The arithmetic also includes the bottom and two hidden sides.

16top faces
16bottom faces
10each side profile
4side directions
Top + bottom + four sides
16 + 16 + 4×10 = 72

Why does one side profile have 10 faces?

1+2+3+4 = 10
Reasonableness check: 30 separate cubes would have 180 faces. Joining them must reduce the surface area, so 72 is plausible.

Complete the directional count

Mission 6

Count a solid with a tunnel: whole minus removed cubes

Our one-tunnel model is a 4×5×4 cuboid: 4 units wide, 5 units high, and 4 units deep. A rectangular tunnel 2 units wide and 1 unit high runs through all 4 units of depth.

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Hollow cuboidUnit cubes

Volume ledger

Complete cuboid:

4×5×4 = 80 cubes

Tunnel:

2×1×4 = 8 cubes removed

Wooden cubes remaining:

80−8 = 72 cubes
Important: The tunnel length is the cuboid’s full depth. A hole visible on the front represents missing cubes all the way through.

Use whole minus hole

Mission 7

Paint the outside, then inspect the unit cubes

A 5×3×4 cuboid is painted on every outer face and cut into 60 unit cubes. All dimensions are in units. The cuboid is painted before cutting; the new cut faces receive no paint. A cube’s location determines how many painted faces it has.

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Horizontal sliceLayer 1 of 4
Layer 1
3 painted2 painted1 painted0 painted

Each square represents one unit cube in the selected horizontal layer.

Three painted faces

Only the 8 corner cubes.

8

Two painted faces

Edge cubes, excluding corners.

24

One painted face

Face-interior cubes.

22

No painted faces

Interior cubes:

(5−2)(3−2)(4−2)=6
Check the partition: 8+24+22+6=60, exactly the cuboid’s volume.
Why are the middle counts 24 and 22?

For exactly two painted faces, count cubes along edges, leaving out the corners. Four edges have 5−2=3 middle cubes; four have 3−2=1; four have 4−2=2. Total: 4×3+4×1+4×2=24.

For exactly one painted face, count the inside of each outer face, leaving out its edges. Two 5×3 faces contribute 2×3×1=6; two 5×4 faces contribute 2×3×2=12; two 3×4 faces contribute 2×1×2=4. Total: 6+12+4=22.

Use the layer slider to find examples of each type. The number printed on each cell is its painted-face count.

Classify the painted cubes

Mission 8

The same volume can have different surface areas

Rearranging cubes does not change how many cubes there are, but it changes how many faces touch neighbours.

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Arrangement A: compact slab4 cubes
Surface area = 16
Arrangement B: long rod4 cubes
Surface area = 18
Compact solids hide more faces. Both arrangements have volume 4 cubic units, but the slab has more contacts and therefore less surface area.

Compare without mixing the quantities

Mission 9

Unit-cube workshop

Solve at least six of the eight problems correctly. Give areas in square units and volumes in cubic units, except the Mission 2 staircase uses cm² and cm³. Use a layer count for volume and a contact or directional count for surface area.

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1. A 2×3×4 cuboid contains how many unit cubes?

2. Layers contain 9, 4, and 1 cubes. What is the total volume?

3. Five unit cubes form one straight row. Find the surface area.

4. A 2×2×2 cube has what surface area?

5. The four-layer staircase in Mission 2 has what volume?

6. The Mission 2 staircase has what surface area (cm²)?

7. How many cubes remain in the tunnel model in Mission 6?

8. How many cubes are completely unpainted in the 5×3×4 painted cuboid?

Mission 10

Exit ticket

Answer all five questions correctly to earn the Unit-Cube Spatial Navigator certificate.

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Teaching notes

The chapter introduction recommends building or manipulating physical models.

The height-map model, contact-pair visual, same-volume comparison, original-independent explanations, diagnostic items, and exit ticket are additional instructional scaffolds. All diagrams are redrawn and are not necessarily to scale.