The cube and the box

Geometry, grade 7 Lesson 15 of 18Learning

The cube and the box

A solid

A triangle, a rectangle and a circle are shapes in the plane: they have a length and a width, and are flat like a sheet of paper. A solid is a shape in space: it has a height too, and takes up room.

A shoebox, a die, a wardrobe and a room are solids.
A ball and a drink can are solids too, but they have curved surfaces.
The box and the cube are solids whose surfaces are all flat.

The box

A box (a cuboid) is a solid like a shoebox, a milk carton or a brick.

Faces: a box has 6 faces, all rectangles. Every two faces opposite each other are congruent.
Edges: an edge is a segment where two faces meet. A box has 12 edges.
Vertices: a vertex is a point where three edges meet. A box has 8 vertices.
Three dimensions: the edges fall into three groups of four equal edges – the length $\displaystyle a$ , the width $\displaystyle b$ and the height $\displaystyle c$ .

The cube

A cube is a box whose edges are all equal, like a die or a sugar cube.

Its 6 faces are all congruent squares.
It has 12 edges and 8 vertices too.
Its length, width and height are equal: $\displaystyle a$ .

Exercise 1

A box is 8 cm long, 5 cm wide and 3 cm high.

Write the measurements of each of its six faces.

The box’s frame is built of iron wire – a wire along every edge. How many cm of wire does it take?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Drawing a box

On flat paper a box is drawn so that it looks like a solid. Only three of its faces show: the front, the top and a side. The edges at the back, which cannot be seen, are drawn dashed.

Draw the front face: a rectangle.
Draw the back face: an equal rectangle, moved a little right and up.
Join each vertex of the front face to the vertex behind it. The three edges meeting at the hidden vertex – bottom left at the back – are dashed.

The edges going backwards are drawn shorter and slanting, so the top and side faces look like parallelograms. On the box itself they are rectangles.

Exercise 2

Draw a cube whose edge is 3 squares long.
Draw the hidden edges dashed.

A net

Cut a cardboard box along some of its edges and spread it flat on the table. What you get is a net of the box: all six faces, joined to each other along their sides.

A net of a cube is six squares. But not every six joined squares are a net of a cube:

The cross folds into a cube: the four squares in the row go round the cube, 1 is the lid and 6 the base.
The rectangle does not fold into a cube: folding up its top row, two squares land on each other, and one face stays open.
A cube has exactly 11 different nets.

Exercise 3

Which of the shapes are nets of a cube?
Imagine folding each one (or cut it out of paper and try), and explain.

Opposite faces in a net

On a cube every face is opposite one other face – its opposite face. In the cross net above:

In the row of squares 2, 3, 4, 5, the faces with one square between them are opposite: 2 and 4, 3 and 5.
The square above the row and the square below it are opposite: 1 and 6.
Two faces that touch in the net are never opposite.

Exercise 4

On a die, the numbers on every two opposite faces add up to 7.
Here is a net of a die. Three faces have their numbers, and three are marked $\displaystyle A$ , $\displaystyle B$ , $\displaystyle C$ .
Which number must be on each of them? Explain.

\displaystyle A

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle B

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle C

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The net of a box

A box has a net too: six rectangles, three pairs of congruent rectangles. Here is a net of a box 4 cm long, 3 cm wide and 2 cm high:

Two rectangles of 4 by 3: the base and the lid.
Two rectangles of 4 by 2: the front and back faces.
Two rectangles of 3 by 2: the two side faces.
Two sides that meet when it is folded are the same edge, so they are equally long.

Exercise 5

Draw a net of a box 4 cm long, 3 cm wide and 1 cm high.
Each square is 1 cm.

Formulas: geometry, grade 7

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Angles

The sum of angles – the ray $\displaystyle BD$ runs inside $\displaystyle \angle ABC$
$\displaystyle \angle ABC$ $\displaystyle =$ $\displaystyle \angle ABD+\angle DBC$
The difference of angles
$\displaystyle \angle DBC$ $\displaystyle =$ $\displaystyle \angle ABC-\angle ABD$
The bisector $\displaystyle BD$ cuts $\displaystyle \angle ABC$ into two equal angles
$\displaystyle \angle ABD=\angle DBC=\frac{\angle ABC}{2}$
Adjacent angles
$\displaystyle \alpha+\beta=180^{\circ}$
Vertical angles are equal
$\displaystyle \alpha=\gamma$
The angles around a point
$\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}$

The angles of a triangle and a quadrilateral

The angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
The two acute angles of a right triangle
$\displaystyle \alpha+\beta=90^{\circ}$
In an isosceles triangle the base angles are equal
$\displaystyle \angle B=\angle C$ $\displaystyle AB=AC$
In an equilateral triangle all the angles are equal
$\displaystyle \alpha=\beta=\gamma=60^{\circ}$
The angles of a quadrilateral
$\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ}$

The perimeter and area of a rectangle and a square

The area of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot b$
The perimeter of a rectangle
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2\cdot(a+b)$
The area of a square – $\displaystyle a$ the side
$\displaystyle S$ $\displaystyle =$ $\displaystyle a^2$
The perimeter of a square
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4\cdot a$

The areas of polygons

The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a right triangle – $\displaystyle a$ , $\displaystyle b$ the legs
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot b}{2}$ $\displaystyle \angle C=90^{\circ}$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$

The box and the cube

The surface area of a box – $\displaystyle a$ the length, $\displaystyle b$ the width, $\displaystyle c$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle 2(ab+bc+ac)$
The surface area of a cube – $\displaystyle a$ the edge
$\displaystyle S$ $\displaystyle =$ $\displaystyle 6a^2$
The volume of a box
$\displaystyle V$ $\displaystyle =$ $\displaystyle a\cdot b\cdot c$
The volume of a cube
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$