Spatial vision: views and building maps

The plane and space, grade 12 (3 units) Lesson 11 of 15Learning

Spatial vision: views and building maps

A building of cubes

This is a building of equal cubes, standing on a table:

The drawing shows three faces of each cube: the front, the right side and the top.
Some cubes are hidden behind other cubes: the drawing alone does not always tell how many cubes there are.

Three views

A view is what one sees looking at the building straight from one side, with one eye: a square for each cube seen, with no depth. These are the building’s three views:

The view from the front: in each column one sees only the tallest of everything standing behind it – 2, 3, 1.
The view from the right: someone standing to the right of the building sees its front at their left. In each row (from the front backwards) one sees the tallest – 1, 3, 2.
The view from above: the building’s floor plan – a square for each place a cube stands on, the front at the bottom.

The building map

A building map is the view from above, with the number of cubes standing on each square, one on another, written in it. The map describes the building exactly, and answers everything:

The number of cubes – the sum of all the numbers: $\displaystyle {1+2+2+3+1+1+1+1}={12}$ .
The view from the front – in each column of the map, the greatest number.
The view from the right – in each row of the map, the greatest number; the front row at the left of the view.

Exercise 1

This is the building map of another building.

How many cubes does the building have?

/ – 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

Draw the view from the front and the view from the right.

Exercise 2

This is another building of cubes:

Which of the views is the view from the right? What are the other two? Explain.

The number of cubes from three views

Three views do not always settle how many cubes there are. The greatest number of cubes that fits them is found like this:

On each square of the view from above, cubes can stand up to the height of its column in the view from the front, and also up to the height of its row in the view from the right.
So in each square write the smaller of the two heights, and add.

For the first building’s views: in the back row this gives $\displaystyle 2$ and $\displaystyle 2$ (the building has 1 and 2 there), and in the other squares exactly what there is. These views fit the building of 12 cubes, and also a building of $\displaystyle {12+1}={13}$ cubes.

Exercise 3

These are the views of a building: 1 from the front, 2 from the right, 3 from above.
What is the greatest number of cubes the building can have?

/ – 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

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 $\displaystyle c$
$\displaystyle p^2$ $\displaystyle =$ $\displaystyle (a+b)^2+c^2$