---
title: "Cubes and Boxes"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 35
exercises: 10
source: https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes
---

# Chapter 35 — Cubes and Boxes

Leaving the flat page: [solids](#def-g5-solids-def) are the shapes of the real world — dice, cereal boxes, cans, balls. This chapter learns to name them, count their [faces](#def-g5-solids-def) and [edges](#def-g5-solids-def), unfold them into [nets](#def-g5-solids-net), and measure their [volume](#def-g5-solids-volume) by counting little [cubes](#def-g5-solids-def). ([Volume](#def-g5-solids-volume) formulas come in [Chapter 43](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#ch-g6-measure).)

## 35.1 The solid family

**Definition 35.1 (Solids and their vocabulary).**

A *solid* occupies space. Flat sides are *faces*; faces meet along *edges*; edges meet at *vertices*. The family portraits:

- the *cube* : $6$ square faces, $12$ edges, $8$ vertices;
- the *box* (rectangular prism): $6$ rectangular faces, $12$ edges, $8$ vertices;
- the *cylinder* : two disk faces and one rolled face (a can);
- the *pyramid* : a [polygon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) base and triangles meeting at an apex;
- the *ball* (sphere): one perfectly round surface, no face, no edge.

![The solid gallery. Dashed lines show hidden edges — the back of the solid, seen through it.](https://one-course.com/images/onecourse/chapters/math-1/g5-solids/fig-483760da7e02.svg)

*The [solid](#def-g5-solids-def) gallery. Dashed lines show hidden [edges](#def-g5-solids-def) — the back of the [solid](#def-g5-solids-def), seen through it.*

**Example 35.2 (Counting on the cube).**

Check on a die: $6$ [faces](#def-g5-solids-def) (the numbers $1$ to $6$), $8$ [vertices](#def-g5-solids-def) (the corners), $12$ [edges](#def-g5-solids-def) (run a finger along them: $4$ on top, $4$ at the bottom, $4$ vertical). And $6 + 8 = 12 + 2$ — the same curious pattern holds for the [box](#def-g5-solids-def) and the [pyramid](#def-g5-solids-def) ([Example 62.2](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#ex-g8-solids-count) returns to it).

## 35.2 Nets

**Definition 35.3 (Net).**

A *net* of a [solid](#def-g5-solids-def) is a flat drawing of all its [faces](#def-g5-solids-def), attached along [edges](#def-g5-solids-def), that folds up into the [solid](#def-g5-solids-def) — the [solid](#def-g5-solids-def) unwrapped like a cardboard [box](#def-g5-solids-def) flattened for recycling.

![Three arrangements of six squares. The first two fold into a cube; the third does not — when folding, two squares land on the same face and the cube stays open. Cut them out and try!](https://one-course.com/images/onecourse/chapters/math-1/g5-solids/fig-f6adbaf2d870.svg)

*Three arrangements of six squares. The first two fold into a [cube](#def-g5-solids-def); the third does not — when folding, two squares land on the same [face](#def-g5-solids-def) and the [cube](#def-g5-solids-def) stays open. Cut them out and try!*

**Example 35.4 (Reading a net).**

On a die, opposite [faces](#def-g5-solids-def) add up to $7$. On a [net](#def-g5-solids-net), opposite [faces](#def-g5-solids-def) are the ones separated by exactly one square in a row (or around a corner) — not the neighbors! Marking the pairs on the [net](#def-g5-solids-net) before folding is a great exercise in seeing space flat.

## 35.3 Volume: counting cubes

**Definition 35.5 (Volume by counting).**

The *volume* of a [solid](#def-g5-solids-def) built from unit [cubes](#def-g5-solids-def) is the number of [cubes](#def-g5-solids-def) it contains. The unit can be the centimeter [cube](#def-g5-solids-def) (cm$^3$) or the meter [cube](#def-g5-solids-def) (m$^3$).

**Example 35.6 (Counting a box layer by layer).**

A [box](#def-g5-solids-def) $4$ [cubes](#def-g5-solids-def) long, $3$ wide, $2$ high: the bottom layer holds $4 \times 3 = 12$ [cubes](#def-g5-solids-def), and there are $2$ layers:

$$
12 \times 2 = 24 \text{ cubes} .
$$

Counting by layers always works — and it quietly proves the formula $V = L \times w \times h$ of [Chapter 43](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#ch-g6-measure).

![A 4 × 3 × 2 box of unit cubes, counted layer by layer.](https://one-course.com/images/onecourse/chapters/math-1/g5-solids/fig-dd505635813d.svg)

*A $4 \times 3 \times 2$ [box](#def-g5-solids-def) of unit [cubes](#def-g5-solids-def), counted layer by layer.*

**Example 35.7 (Same cubes, different shapes).**

Eight unit [cubes](#def-g5-solids-def) can build a $2 \times 2 \times 2$ [cube](#def-g5-solids-def), a $8 \times 1 \times 1$ stick, or a $4 \times 2 \times 1$ slab: three different [solids](#def-g5-solids-def), one [volume](#def-g5-solids-volume) ($8$ [cubes](#def-g5-solids-def)). [Volume](#def-g5-solids-volume) counts the material, not the shape.

## 35.4 Exercises

**Exercise 35.1 ★.**

Name a real object shaped like: a [cube](#def-g5-solids-def); a [box](#def-g5-solids-def); a [cylinder](#def-g5-solids-def); a ball; a [pyramid](#def-g5-solids-def).

**Solution of Exercise 35.1.**

For instance: a die ([cube](#def-g5-solids-def)); a cereal [box](#def-g5-solids-def) ([box](#def-g5-solids-def)); a can ([cylinder](#def-g5-solids-def)); a football (ball); certain roofs or the monuments of Egypt ([pyramid](#def-g5-solids-def)).

**Exercise 35.2 ★.**

How many [faces](#def-g5-solids-def), [edges](#def-g5-solids-def) and [vertices](#def-g5-solids-def) has a [box](#def-g5-solids-def)? Check the pattern [faces](#def-g5-solids-def) $+$ [vertices](#def-g5-solids-def) $=$ [edges](#def-g5-solids-def) $+ 2$.

**Solution of Exercise 35.2.**

A [box](#def-g5-solids-def) has $6$ [faces](#def-g5-solids-def), $12$ [edges](#def-g5-solids-def), $8$ [vertices](#def-g5-solids-def) — and $6 + 8 = 12 + 2$. ✓

**Exercise 35.3 ★.**

Which [faces](#def-g5-solids-def) of a [box](#def-g5-solids-def) are identical? (Group the six [faces](#def-g5-solids-def) in pairs.) What is special about the [cube](#def-g5-solids-def)’s [faces](#def-g5-solids-def)?

**Solution of Exercise 35.3.**

The six [faces](#def-g5-solids-def) come in three pairs of identical opposite [faces](#def-g5-solids-def) (top–bottom, front–back, left–right). On a [cube](#def-g5-solids-def), all six [faces](#def-g5-solids-def) are identical squares.

**Exercise 35.4 ★.**

Draw a [net](#def-g5-solids-net) of a [cube](#def-g5-solids-def) of side $2$ cm (use the cross-shaped model of the chapter), cut it out and fold it.

**Solution of Exercise 35.4.**

Cross-shaped [net](#def-g5-solids-net): four $2$ cm squares in a row, one above and one below the second square. Folding the ring of four makes the sides; the two extra squares close the top and bottom.

**Exercise 35.5 ★.**

On the cross-shaped [net](#def-g5-solids-net) of a die, place the numbers $1$ to $6$ so that opposite [faces](#def-g5-solids-def) sum to $7$. (Use [Example 35.4](#ex-g5-solids-read) to find the opposite pairs first.)

**Solution of Exercise 35.5.**

On the cross [net](#def-g5-solids-net) (four squares in a row, one above, one below the second), the opposite pairs are: squares $1$ and $3$ of the row; squares $2$ and $4$ of the row; the top and bottom squares. One correct labelling: row $= 1, 2, 6, 5$; top $= 3$; bottom $= 4$ — then $1{+}6$, $2{+}5$, $3{+}4$ all make $7$.

**Exercise 35.6 ★.**

Count the unit [cubes](#def-g5-solids-def): a [box](#def-g5-solids-def) $5$ long, $2$ wide, $3$ high; a [cube](#def-g5-solids-def) of side $3$; an L-shaped stack made of a $3 \times 2 \times 1$ slab with a $1 \times 2 \times 1$ slab on top.

**Solution of Exercise 35.6.**

[Box](#def-g5-solids-def): $5 \times 2 = 10$ per layer, $3$ layers: $30$ [cubes](#def-g5-solids-def). [Cube](#def-g5-solids-def): $3 \times 3 \times 3 = 27$. L-stack: $3 \times 2 \times 1 = 6$ plus $1 \times 2 \times 1 = 2$: $8$ [cubes](#def-g5-solids-def).

**Exercise 35.7 ★.**

A [box](#def-g5-solids-def) holds exactly $12$ unit [cubes](#def-g5-solids-def) in one layer. Give two possible layer shapes ($L \times w$), and for each, the [box](#def-g5-solids-def)’s [volume](#def-g5-solids-volume) if it has $3$ such layers.

**Solution of Exercise 35.7.**

Layers of $12$: for instance $4 \times 3$ or $6 \times 2$ (or $12 \times 1$). With $3$ layers, the [volume](#def-g5-solids-volume) is $12 \times 3 = 36$ [cubes](#def-g5-solids-def) in every case.

**Exercise 35.8 ★.**

With $27$ unit [cubes](#def-g5-solids-def), what [cube](#def-g5-solids-def) can you build? With $64$? (Think of the layer counting backwards.)

**Solution of Exercise 35.8.**

$27 = 3 \times 3 \times 3$: a [cube](#def-g5-solids-def) of side $3$. $64 = 4 \times 4 \times 4$: a [cube](#def-g5-solids-def) of side $4$.

**Exercise 35.9 ★★.**

Sugar [cubes](#def-g5-solids-def) of side $1$ cm come in a [box](#def-g5-solids-def) $10$ cm long, $5$ cm wide, $4$ cm high (interior sizes). How many sugar [cubes](#def-g5-solids-def) fit? If a family uses $6$ [cubes](#def-g5-solids-def) a day, for how many days does one [box](#def-g5-solids-def) last ([round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) sensibly)?

**Solution of Exercise 35.9.**

The [box](#def-g5-solids-def) holds $10 \times 5 \times 4 = 200$ sugar [cubes](#def-g5-solids-def). At $6$ per day: $200 = 6 \times 33 + 2$, so the [box](#def-g5-solids-def) lasts $33$ full days (and $2$ [cubes](#def-g5-solids-def) remain for the morning of day $34$).

**Exercise 35.10 ★★.**

A $3 \times 3 \times 3$ [cube](#def-g5-solids-def) is painted red on the outside, then cut into $27$ unit [cubes](#def-g5-solids-def). How many small [cubes](#def-g5-solids-def) have paint on exactly $3$ [faces](#def-g5-solids-def)? On $2$? On $1$? On none? (Check: the four counts must add up to $27$.)

**Solution of Exercise 35.10.**

Paint on $3$ [faces](#def-g5-solids-def): the $8$ corner [cubes](#def-g5-solids-def). On $2$ [faces](#def-g5-solids-def): the middles of the $12$ [edges](#def-g5-solids-def): $12$ [cubes](#def-g5-solids-def). On $1$ [face](#def-g5-solids-def): the centers of the $6$ [faces](#def-g5-solids-def): $6$ [cubes](#def-g5-solids-def). On none: the single hidden center [cube](#def-g5-solids-def): $1$. Check: $8 + 12 + 6 + 1 = 27$. ✓