Constructions and Tilings

Why This Matters

Look at the floor of a railway station. Or the wall of a bathroom. Or the bricks in a wall. You will see the same shape, copied again and again, fitting together with no gaps.

Now look at a building with arches, or a window with a neat design carved in stone. Someone had to draw those shapes first. They had to be exact. A pillar that is even a little crooked looks wrong.

How do people draw shapes so exactly? Not by guessing. They use two simple tools: a ruler and a compass. With just these two, you can draw perfectly straight lines, perfect circles, exact angles, and exact triangles.

This chapter has two big parts.

First, constructions — drawing exact shapes with a ruler and compass.

Second, tilings — fitting shapes together to cover a flat surface with no gaps and no overlaps, like floor tiles. You will also learn the surprising reason why some shapes tile and others simply cannot.

The best part is this: you will not just learn the steps. You will learn why each step works. Once you know why, you never have to memorise the steps. You can work them out yourself.

The Big Idea

With only an unmarked ruler and a compass, you can build exact shapes — not by measuring with numbers, but by copying equal lengths and equal angles. The compass keeps a length fixed, so it copies distances perfectly; clever use of equal arcs lets you copy an angle and draw parallel lines and triangles. And when you fit shapes together to cover a flat surface, one rule decides everything: the angles meeting at each point must add up to exactly 360°, a full turn. That single rule explains why squares, triangles and hexagons tile a floor, but a regular pentagon never can.

Let’s Break It Down

Before we draw anything, let us refresh two ideas from earlier work. The whole chapter leans on them.

Your tools: ruler and compass

You only need two tools for this chapter.

A ruler is a straight edge. We use it to draw straight lines and to join two points. Here is the surprising part: for most constructions we use the ruler without looking at the cm marks at all. We only use its straight edge. A ruler used this way is called an unmarked ruler.

A compass is the two-legged tool. One leg has a sharp point; the other holds a pencil. You open it to some width, press the point down, and swing the pencil leg to draw a circle or part of a circle.

A part of a circle is called an arc. For example, if you only swing the pencil a little way, you draw a short curved line — that is an arc.

Here is the one magic property of the compass that makes everything work:

A compass keeps the same width until you change it.

So if you open it to the length of one line and carry it across the page, you can mark that exact same length somewhere else — without ever reading a number.

This is why we don’t need the cm marks. The compass copies a length perfectly. A ruler with marks can only measure to the nearest millimetre, and your eye can misread it. The compass makes no such error. That is why a compass construction is more exact than measuring.

Why does a compass draw a perfect circle? Because every point on the circle is the same distance from the sharp point in the middle. That distance is the width you set — the radius. The compass cannot change that distance while you swing it, so every point lands exactly the same distance away. A perfect circle is just “all the points at one fixed distance from a centre”.

Constructing a line parallel to a given line through a point

Here is our first real construction. Suppose someone draws a line m on the page and marks a point B somewhere above it. The task is: draw a line through B that is parallel to m.

We cannot just slide a ruler “by eye” and hope it stays parallel. We need a method that is exact.

Remember the recap: if a transversal crosses two lines and makes equal corresponding angles, the two lines are parallel. So our plan is:

1. Draw any slanting line through B that also cuts m. This is our transversal. Call it l.
2. Look at the angle l makes with m.
3. At B, draw a new line that makes the same angle with l.

Because the angles match, the new line is parallel to m. We just need a compass way to copy an angle exactly. We do that by copying the little “gap” of an arc. Figure 6.1 shows all four steps.

Tap to zoom

Let us walk through Figure 6.1 slowly.

In Step 1, we have the line m, the slanting line l crossing it at A, and our point B sitting on l.

In Step 2, we put the compass point at A and draw one arc. It crosses l at D and crosses m at C. Notice what this arc has done: the two points C and D, together with A, record the angle at A. The distance AC and AD are fixed (they are the radius), and the gap CD depends only on how wide the angle is.

In Step 3, without changing the compass width at all, we move the point to B and draw the same arc. It crosses l at E.

In Step 4, we measure the gap CD with the compass and mark the same gap from E to get F. Now the angle at B (between l and the line BF) is an exact copy of the angle at A. We draw the line n through B and F.

Why does this work? This is the important part — not just that it works, but why. By copying the radius and copying the gap CD, we built an angle at B that is exactly equal to the angle at A. Those are corresponding angles for the transversal l. Equal corresponding angles mean the two lines are parallel. So n ∥ m. We never measured a single number. We just copied a length and a gap with the compass, and geometry did the rest.

Constructing triangles — SSS, SAS, ASA

Now we build triangles. A triangle has three sides and three angles — six measurements in all. Here is something surprising: you do not need all six to fix a triangle. Just the right three are enough. The triangle then has only one possible shape and size.

There are three handy sets of “three things” that fix a triangle. Their names are short codes:

• SSS = Side, Side, Side. You know all three sides.
• SAS = Side, Angle, Side. You know two sides and the angle between them.
• ASA = Angle, Side, Angle. You know two angles and the side between them.

Let us see each one.

SSS — three sides given. Say the sides are 6 cm, 5 cm and 4 cm. The trick is the compass. You draw the longest side as a base. Then, since the third corner must be a fixed distance from each end, you draw an arc from each end. Where the two arcs cross is the only place the third corner can sit. Figure 6.2 shows this.

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In Figure 6.2, notice the heart of the method in Step 2. The point C must be 4 cm from A — so it lies somewhere on the arc of radius 4 cm around A. It must also be 5 cm from B — so it lies on the arc of radius 5 cm around B. The only point on both arcs is where they cross. That fixes C completely. This is why three sides give exactly one triangle.

Let us build one carefully, step by step.

SAS — two sides and the angle between them. Here you know two sides and the angle that sits between those two sides. For example: AB = 6 cm, the angle at A = 50°, and AC = 4 cm. You draw the base AB, draw the 50° angle at A (using the compass — we will see in the hexagon section how to build exact angles), mark 4 cm along that new arm to fix C, then join BC. Panel (a) of Figure 6.3 shows it.

ASA — two angles and the side between them. Here you know one side and the two angles at its ends. For example: PQ = 6 cm, the angle at P = 50°, and the angle at Q = 40°. You draw the side PQ, draw the 50° arm at P and the 40° arm at Q, and where the two arms cross is the third corner R. Panel (b) of Figure 6.3 shows it.

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Look at Figure 6.3. In SAS (panel a), the angle is trapped between the two known sides — that is what “the included angle” means. In ASA (panel b), the side is trapped between the two known angles. As long as the given part is in the right place, the triangle is fixed and has only one shape.

Here is a useful comparison so you remember which is which.

Tilings — shapes that cover a plane with no gaps

Now for the second half of the chapter. Look at a tiled floor. The tiles fit together perfectly. No gaps where dust could collect, and no tile sitting on top of another.

Covering a flat surface using copies of a shape (or shapes), with no gaps and no overlaps, is called a tiling. Another name for it is a tessellation.

A “flat surface” that goes on forever in every direction is called a plane. So “tiling the plane” means covering an endless flat sheet, with the pattern repeating forever.

You already know some shapes tile. Squares obviously do — that is your bathroom floor. So do rectangles, like bricks in a wall. What about other shapes?

It turns out equilateral triangles tile too (point them up and down, up and down, in rows). And so do regular hexagons — six-sided shapes with all sides and angles equal. A honeycomb made by bees is exactly this: hexagons fitting together with no wasted space. Figure 6.4 shows all three.

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A neat tiling fact to test on small grids: take a grid of unit squares, like a 4 × 6 grid (4 rows, 6 columns, so 24 small squares). Can you cover it with 2 × 1 tiles (each tile covers two squares)? Yes — each tile covers 2 squares, and 24 is even, so it can work.

But a 5 × 7 grid has 35 small squares. Each 2 × 1 tile covers 2 squares. You can only ever cover an even number of squares with these tiles (2, 4, 6, …). Since 35 is odd, no arrangement can ever cover it exactly. One square will always be left over. This is a lovely example of proving a tiling is impossible, just by counting — without trying every arrangement.

Which shapes tile, and WHY

Here is the deepest idea in the chapter. Why do squares, triangles and hexagons tile, but a regular pentagon does not?

The answer is the 360° rule. At every point where corners of tiles meet, the angles of those corners must add up to exactly 360° — a full turn. If they add to less than 360°, there is a leftover gap. If they add to more than 360°, the tiles overlap. Only an exact 360° gives a perfect fit.

Let us test the shapes.

• A square corner is 90°. Four squares meet at a point: 90° + 90° + 90° + 90° = 360°. Perfect fit. ✓
• An equilateral triangle corner is 60°. Six of them meet at a point: 60° × 6 = 360°. Perfect fit. ✓
• A regular hexagon corner is 120°. Three of them meet at a point: 120° × 3 = 360°. Perfect fit. ✓

Figure 6.5 shows the hexagon case clearly: three hexagon corners, each 120°, meeting at one point and filling the full turn exactly.

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Now the surprise. A regular pentagon (five equal sides, five equal angles) does not tile the plane. Why not? Because of its corner angle. Each corner of a regular pentagon is 108°. Let us try to fit pentagons around a point:

• Two pentagons: 108° + 108° = 216°. Still lots of room left.
• Three pentagons: 108° + 108° + 108° = 324°. Close to 360°, but not quite.

We are now at 324°, and a full turn is 360°. The gap left is 360° − 324° = 36°. A fourth pentagon needs 108° of room, but only 36° is left. It cannot fit. So three pentagons leave a 36° wedge of empty space, and you cannot fill it. Figure 6.6 shows this gap.

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So the real test is simple: does the shape’s corner angle divide evenly into 360°?

• 90° goes into 360° exactly 4 times → square tiles.
• 60° goes into 360° exactly 6 times → triangle tiles.
• 120° goes into 360° exactly 3 times → hexagon tiles.
• 108° does not divide 360° evenly (360 ÷ 108 is about 3.33) → pentagon does not tile.

That is the whole secret. A picture is worth remembering here: think of the angles meeting at a point as slices that must add up to one full circle. If the slice size fits a whole number of times into the circle, the shape tiles. If not, you are always left with an awkward gap.

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Summary

• A construction means drawing an exact shape using only a ruler (straight edge) and a compass — usually without reading the cm marks, because the compass copies lengths perfectly.
• To draw a line through a point parallel to a given line, you copy the angle the transversal makes, using equal arcs. Equal corresponding angles force the lines to be parallel (Figure 6.1).
• A triangle is fixed by the right three measurements: SSS (three sides), SAS (two sides and the angle between them), or ASA (two angles and the side between them).
• In SSS, the third corner is found where two arcs (one from each end of the base) cross — that point is the only one at both correct distances (Figure 6.2).
• A tiling (or tessellation) covers a flat surface with copies of a shape, leaving no gaps and no overlaps.
• The key rule: at every meeting point, the corner angles must add up to exactly 360° (a full turn).
• Squares (90°), equilateral triangles (60°), and regular hexagons (120°) tile because their corner angle divides 360° evenly (Figures 6.4, 14.5).
• A regular pentagon (108°) cannot tile alone, because 108° does not divide 360° evenly — three corners leave a 36° gap (Figure 6.6).

What’s Next

You can now build exact shapes and explain why some shapes tile and others don’t. The next chapter, Finding the Unknown, turns from drawing to solving. You will treat an unknown number like a hidden box and work out what must be inside it — the start of real algebra. The same careful, step-by-step thinking you used here for constructions will help you there too: every step has a reason, and nothing is just a given.