Nets and the shortest path on a solid

The plane and space, grade 12 (3 units) Lesson 12 of 15Learning

Nets and the shortest path on a solid

The net of a cube

The net of a solid is what one gets by cutting it along some of its edges and laying it flat. Folding the net back gives the solid.

A cube has 11 different nets, all made of six squares. Not every six joined squares are a net: in some shapes two faces fall on each other when folded.
Opposite faces are two faces facing each other on the cube. In a row of squares in a net, a square and the square after the next are opposite faces: in the net in the drawing $\displaystyle A$ is opposite $\displaystyle C$ , $\displaystyle B$ opposite $\displaystyle D$ , and $\displaystyle E$ opposite $\displaystyle F$ .
On a die the sum of any two opposite faces is 7: 1 opposite 6, 2 opposite 5, 3 opposite 4.

Exercise 1

Which of the shapes are nets of a cube? Imagine folding each shape, and explain.

Exercise 2

Which of the nets folds into a die – the sum of any two opposite faces 7? Explain.

Nets of a cylinder and a cone

The net of a cylinder: a rectangle and two discs. The rectangle’s length is the circle’s circumference, $\displaystyle 2\pi r$ – it wraps round it exactly once.
The net of a cone: a disc, and a sector of a bigger disc whose radius is the slant height $\displaystyle l$ . The sector’s arc wraps round the base, so its length is $\displaystyle 2\pi r$ .

Exercise 3

The net of a can that is a cylinder is cut from tin: the radius is 5 cm, and the height 12 cm.
How long is the rectangle in the net? Round to two decimal places.

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The shortest path on the surface of a box

An ant walks on the surface of a box of 4 by 3, 2 high, from the corner $\displaystyle A$ at the bottom to the corner $\displaystyle B$ farthest from it at the top. It cannot go through the inside of the box.

On a flat sheet, the shortest way between two points is a straight segment. So unfold the faces the ant walks on into one sheet – part of a net – and draw a straight line from $\displaystyle A$ to $\displaystyle B$ .
Here it walks across the front and the lid. Unfolded, they are a rectangle of 4 by $\displaystyle {2+3}={5}$ , and the path is its diagonal:

\sqrt{4^2+5^2}=\sqrt{41}\approx6.40

There are two more ways to unfold: a rectangle of 3 by $\displaystyle {4+2}={6}$ , and a rectangle of 2 by $\displaystyle {4+3}={7}$ . Check them all and choose the shortest:

$\displaystyle {\sqrt{3^2+6^2}}={\sqrt{45}\approx6.71}$
$\displaystyle {\sqrt{2^2+7^2}}={\sqrt{53}\approx7.28}$

The shortest path is $\displaystyle \sqrt{41}\approx6.40$ – when the longest edge, 4, stays alone, and the two short ones are joined.

\displaystyle p^2

\displaystyle =

\displaystyle (a+b)^2+c^2

Exercise 4

A fly walks on the surface of a cube whose edge is 3 cm, from a vertex to the vertex farthest from it.
How long is the shortest path on the cube’s surface? Round to two decimal places.

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Exercise 5

A room’s floor is a rectangle of 6 m by 4 m, and the room is 3 m high.
A spider is at a corner of the floor, and a fly at the corner of the ceiling farthest from it.
The spider walks on the walls, the floor and the ceiling.
How long is its shortest path? Round to two decimal places.

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Formulas: the plane and space, grade 12 (3 units)

Open as a page

The box and the cube

The volume of a box – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three dimensions
$\displaystyle V$ $\displaystyle =$ $\displaystyle abc$
The surface area of a box – three pairs of equal faces
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2(ab+ac+bc)$
The volume of a cube – $\displaystyle a$ the edge
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$
The surface area of a cube – six square faces
$\displaystyle A$ $\displaystyle =$ $\displaystyle 6a^2$

The prism

The volume of a prism – $\displaystyle B$ the base area, $\displaystyle h$ the height (the distance between the bases)
$\displaystyle V$ $\displaystyle =$ $\displaystyle B\cdot h$
The lateral area – $\displaystyle P$ the perimeter of the base
$\displaystyle L$ $\displaystyle =$ $\displaystyle P\cdot h$
The surface area – the two bases and the lateral surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2B+L$

The cylinder

The volume of a cylinder – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2h$
The curved surface area – a rectangle as long as the base’s circumference
$\displaystyle L$ $\displaystyle =$ $\displaystyle 2\pi rh$
The surface area – the two bases and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2\pi r^2+2\pi rh$

The pyramid and the cone

The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{B\cdot h}{3}$
In a square pyramid – $\displaystyle k$ the height of a side face, $\displaystyle h$ the pyramid’s height, $\displaystyle a$ the side of the base
$\displaystyle k^2$ $\displaystyle =$ $\displaystyle h^2+\left(\frac{a}{2}\right)^2$
The surface area of a square pyramid – the base and four triangles
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2+2ak$
The volume of a cone – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2h}{3}$
The slant height $\displaystyle l$ – the hypotenuse of a triangle whose legs are the radius $\displaystyle r$ and the height $\displaystyle h$
$\displaystyle l^2$ $\displaystyle =$ $\displaystyle r^2+h^2$
The curved surface area of a cone
$\displaystyle L$ $\displaystyle =$ $\displaystyle \pi rl$
The surface area of a cone – the base and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2+\pi rl$

The sphere

The volume of a sphere – $\displaystyle r$ the radius
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4\pi r^3}{3}$
The surface area of a sphere
$\displaystyle A$ $\displaystyle =$ $\displaystyle 4\pi r^2$

Diagonals and distances

The diagonal of a box $\displaystyle d$ – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three dimensions
$\displaystyle d^2$ $\displaystyle =$ $\displaystyle a^2+b^2+c^2$
The diagonal of a cube – $\displaystyle a$ the edge
$\displaystyle d$ $\displaystyle =$ $\displaystyle a\sqrt{3}$
In a square pyramid – $\displaystyle s$ a side edge, $\displaystyle h$ the height, $\displaystyle a$ the side of the base
$\displaystyle s^2$ $\displaystyle =$ $\displaystyle h^2+\frac{a^2}{2}$

Trigonometry in a right-angled triangle

$\displaystyle a$ the leg opposite the angle $\displaystyle \alpha$ , $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
$\displaystyle b$ the leg next to the angle $\displaystyle \alpha$
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
The leg opposite the angle over the leg next to it
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The angle, when both legs are known
$\displaystyle \alpha=\tan^{-1}\left(\frac{a}{b}\right)$

The shortest path on the surface of a box

A path from a vertex to the vertex farthest from it, across two faces: unfolded, the diagonal of a rectangle with sides $\displaystyle a+b$ and $\displaystyle c$
$\displaystyle p^2$ $\displaystyle =$ $\displaystyle (a+b)^2+c^2$