Regular tilings

The plane and space, grade 10 (3 units) Lesson 13 of 15Learning

Regular tilings

A regular polygon

A regular polygon is a polygon whose sides are all equal and whose angles are all equal: an equilateral triangle, a square, a regular pentagon, a regular hexagon …

A regular tiling is a tiling with regular polygons of one kind and one size, meeting vertex to vertex. In the drawings: tilings with triangles, with squares and with hexagons.

The angle of a regular polygon

Diagonals from one vertex cut a polygon with $\displaystyle n$ sides into $\displaystyle n-2$ triangles. In the drawing: a hexagon, $\displaystyle {6-2}={4}$ triangles.

The angles of each triangle add up to $\displaystyle 180^\circ$ , so the sum of the angles of the polygon is:

\displaystyle 180^{\circ}\cdot(n-2)

In a regular polygon all $\displaystyle n$ angles are equal, so each angle is:

\displaystyle \alpha

\displaystyle =

\displaystyle \frac{180^{\circ}\cdot(n-2)}{n}

The polygon	$\displaystyle n$	The angle
equilateral triangle	3	$\displaystyle 60^\circ$
square	4	$\displaystyle 90^\circ$
regular pentagon	5	$\displaystyle 108^\circ$
regular hexagon	6	$\displaystyle 120^\circ$
regular octagon	8	$\displaystyle 135^\circ$

Exercise 1

Calculate the angle of the regular polygon, in degrees.

A pentagon,

\displaystyle {n}={5}

/ – 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

A decagon – 10 sides

/ – 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

A regular polygon with 12 sides

/ – 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 angles around a vertex

In a tiling, the angles meeting at a vertex fill the whole turn around it, with no gap and no overlap: they add up to $\displaystyle 360^\circ$ . So a regular polygon tiles alone only if $\displaystyle 360^\circ$ divides by its angle.

A triangle: $\displaystyle {360\div60}={6}$ – six triangles at each vertex.
A square: $\displaystyle {360\div90}={4}$ .
A hexagon: $\displaystyle {360\div120}={3}$ .
A pentagon: $\displaystyle {360\div108}={3.33\ldots}$ – three pentagons leave a gap of $\displaystyle {360-3\cdot108}={36^\circ}$ , and four have no room. In the drawing: the gap between three pentagons.
A polygon with more than 6 sides: its angle is more than $\displaystyle 120^\circ$ and less than $\displaystyle 180^\circ$ , so three angles are more than $\displaystyle 360^\circ$ and two are less.
So there are only three regular tilings: with triangles, with squares and with hexagons. A beehive is built of hexagons.

Exercise 2

The angle of a regular polygon with 12 sides is $\displaystyle 150^\circ$ .
Can a floor be tiled with such polygons alone? Explain with the angles around a vertex.

A tiling with two polygons

A floor can also be tiled with several kinds of regular polygons together, as long as the angles at each vertex add up to $\displaystyle 360^\circ$ .

In the tiling in the drawing, two octagons and a square meet at each vertex: $\displaystyle {135+135+90}={360}$ .
Such a tiling is common in bathrooms and kitchens.

Exercise 3

At each vertex of a tiling the regular polygons in the note meet. Calculate how many degrees are left for the rest of the vertex.

Two hexagons and two triangles

/ – 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

A 12-sided polygon, and another like it

/ – 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

In part 1 no gap is left – it is a tiling. In part 2 a gap is left: which regular polygon fills it?

Formulas: the plane and space, grade 10 (3 units)

Open as a page

Perimeters

The perimeter of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2(a+b)$
The perimeter of a square – $\displaystyle a$ the side
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4a$
Pi
$\displaystyle \pi\approx3.14$
The circumference – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$

Routes: Pythagoras, distance, speed and time

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2$
The hypotenuse
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^2+b^2}$
A leg
$\displaystyle a$ $\displaystyle =$ $\displaystyle \sqrt{c^2-b^2}$
Speed – $\displaystyle s$ the distance, $\displaystyle t$ the time
$\displaystyle v$ $\displaystyle =$ $\displaystyle \frac{s}{t}$
Distance
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$
Time
$\displaystyle t$ $\displaystyle =$ $\displaystyle \frac{s}{v}$

Areas

The area of a rectangle
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2$
The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a disc – $\displaystyle r$ the radius
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$
A square metre in square centimetres (a dunam is 1000 m²)
$\displaystyle 1\text{ m}^2=100\cdot100\text{ cm}^2=10000\text{ cm}^2$

Tilings

The sum of the angles of a polygon with $\displaystyle n$ sides
$\displaystyle 180^{\circ}\cdot(n-2)$
The angle of a regular polygon with $\displaystyle n$ sides
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \frac{180^{\circ}\cdot(n-2)}{n}$
The angles around a vertex of a tiling
$\displaystyle \alpha_1+\alpha_2+\ldots=360^{\circ}$