Angles, triangles and tessellation

Picture a bathroom floor tiled with squares, or a honeycomb packed with hexagons. Both shapes cover the space with no gaps at all. Now picture a floor tiled with regular pentagons, five equal sides each. Shuffle the pentagons however you like, and a gap opens up that no piece can close. Some shapes fit together perfectly and others never can, and the reason is a number of degrees. It starts with a full turn.

An angle measures an amount of turn, in degrees. All the way round a point is a full turn, three hundred and sixty degrees. Half a turn makes a straight line, one hundred and eighty. A quarter turn is a right angle, ninety degrees, like the corner of a sheet of MDF. The three angles inside any triangle add to one hundred and eighty degrees. Tear the corners off a paper triangle, lay them side by side, and they make a straight line. That is why the total never changes. An equilateral triangle has three sides the same length, and its corners are all the same size too. How big are the angles of an equilateral triangle, in degrees? Sixty degrees each. One hundred and eighty shared equally between three angles is sixty. A right-angled triangle spends ninety of its one hundred and eighty degrees on the right angle. The other two angles share whatever is left between them. A right-angled triangle has an angle of thirty-five degrees. What is the third angle? Fifty-five degrees, because ninety take away thirty-five is fifty-five. Marking out an angle on a part means measuring it with a protractor. A protractor carries two scales that run in opposite directions, one round the inside and one round the outside. Put the centre mark on the corner, and lay the zero line along one arm. Then read the scale that starts at zero on that arm. Here is a corner of a pine shelf bracket, clearly narrower than a right angle. Its other arm crosses the two scales at fifty and at one hundred and thirty. Which reading is the bracket's angle: fifty degrees or one hundred and thirty? Fifty degrees. An angle under ninety takes the smaller reading, and one hundred and thirty belongs to the other scale. Decide first whether the angle is smaller or bigger than a right angle. Then read the scale that agrees with you, and the wrong scale cannot catch you out.

A designer is making wall shelves shaped as regular hexagons, with six equal sides and six equal corners. The shelves clip together in a honeycomb. Draw a line from the centre of the hexagon to each of its six corners. That cuts it into six identical triangles. The six angles at the centre make a full turn. Three hundred and sixty divided by six is sixty, so each centre angle is sixty degrees. In a regular hexagon, each side is as long as the distance from the centre to a corner. So those six triangles are equilateral, and every angle inside them is sixty degrees. Three sixties make one hundred and eighty, which checks. Now point at one corner of the hexagon itself, out on its edge, where the shelf turns. What is the angle at a corner of the hexagon, in degrees? One hundred and twenty degrees. Two triangle corners meet at every corner of the hexagon, and sixty plus sixty is one hundred and twenty. Sixty on its own is one triangle's corner, not the shelf's. Give sixty as the hexagon's corner and you have counted only half of it. The shelves clip together three at a time, all meeting round one point. Do three shelf corners close round one point with no gap, and how do you know? They close perfectly. Three times one hundred and twenty is three hundred and sixty degrees, a full turn, so nothing is left over to make a gap.

Some angles come from sizes rather than shapes. A bird box is one hundred and fifty millimetres wide, and its two roof boards meet at a ridge sixty millimetres above the walls. The roof is a triangle. Split it down the middle and each half is a right-angled triangle, seventy-five millimetres across, which is half of one hundred and fifty, and sixty millimetres high. Draw that half full size. Rule a base line seventy-five millimetres long. At one end, use a set square to stand a line sixty millimetres tall, straight up at ninety degrees. Join the top of that upright line to the far end of the base. That sloping line is the roof board. Put the protractor on the far corner, where the roof board meets the base. A careful drawing measures about thirty-nine degrees, the angle each board makes with the top of the walls. Now the ridge: what angle do the two roof boards make where they meet? About one hundred and two degrees. In the half triangle, the two corners that are not the right angle share ninety. Ninety take away thirty-nine is fifty-one. The ridge holds two of those halves, and two fifty-ones make one hundred and two. Stopping at fifty-one gives only half the ridge. Put the two halves back together before you answer. Without rounding the thirty-nine, the ridge is one hundred and two point seven degrees, so one hundred and three to the nearest degree. A careful drawing lands on one hundred and two or one hundred and three, because a protractor reads to about a degree. One examiner's report, on a question asking for a roof angle, says this: a mixture of mathematical and graphic responses were seen and able to access full marks. Graphic there means drawn, so a careful drawing is a complete answer. The habit that makes it careful is a sharp pencil, the right angle set with a set square, and the protractor read from the zero on the base.

The second route uses a calculator, and it is called the tangent method. Everything in it is named from the angle you want. The side across the triangle from that angle is the opposite. The side touching it, that is not the sloping side, is the adjacent. A wooden doorstop is a wedge. Its base is one hundred and fifty millimetres long, and its square back end is forty millimetres high, so its side is a right-angled triangle. The maker wants the floor angle first, where the sloping face meets the floor. The two short sides are the back and the base. Which side is the opposite for the floor angle: the back or the base? The forty millimetre back. It sits across the triangle from the floor angle. The one hundred and fifty millimetre base touches that angle, so the base is the adjacent. Here is the rule. The tangent of an angle equals the opposite divided by the adjacent. Written down, tan A equals opposite over adjacent, where A is the angle you pointed at. For the doorstop, tan A equals forty over one hundred and fifty. Forty divided by one hundred and fifty is nought point two six six, and the sixes keep going, so keep every digit. That number is the tangent, not the angle. To turn it back into an angle, you use the inverse tangent, written tan to the minus one. Check the calculator is in degrees first: a small D at the top of the display. An R there means radians, and every angle will come out wrong. Press shift, then tan, to get tan to the minus one. Key in forty, divide, one hundred and fifty, close the bracket and press equals. The display reads fourteen point nine three one four one seven one eight. Round once, at the end. Fourteen point nine three is fifteen degrees to the nearest degree, and that is the floor angle. Check it against the sketch. A long, low wedge should meet the floor at a small angle, and fifteen degrees is small, so it fits. Now the thin top edge of the wedge: what is its angle, to the nearest degree? Seventy-five degrees. The two corners that are not the right angle share ninety, and ninety take away fourteen point nine three is seventy-five point nought seven. A different working, for the same corner: tan A equals one hundred and fifty over forty, so A comes out at seventy-five. What has gone wrong in that student's working? The division is upside down. Opposite divided by adjacent means the back divided by the base, and this working divides the base by the back. What they really found is the top edge's angle. From up there, the base is the opposite, and tan to the minus one of three point seven five gives seventy-five point nought seven, the same answer the other route gave. A sketch would have caught it, because a low wedge cannot meet the floor that steeply. That gives two routes to an angle in a right-angled triangle. The tangent method is quick and exact when you know the two shorter sides and have a calculator. A drawing needs only a ruler, a set square and a protractor. Pick the one that suits the question, and check either with a quick sketch.

Tessellation means shapes fitting together edge to edge, with no gaps and no overlaps, like tiles on a floor. A maker laser-cuts coasters from cork sheets, six hundred by four hundred millimetres, and wants the least waste. Wherever corners meet at a point, their angles must add to exactly three hundred and sixty degrees, a full turn. Any less leaves a gap, and any more means the tiles overlap. Squares first. Four corners of ninety degrees meet at a point, and four times ninety is three hundred and sixty, so squares tessellate. The triangle with three equal sides comes after the square. How many sixty degree triangle corners fit round one point, and do they tessellate? Six of them, and yes. Six sixties make three hundred and sixty, a full turn, so equilateral triangles tessellate. Hexagons were settled by the shelves. Three corners of one hundred and twenty make three hundred and sixty, so the honeycomb closes. That leaves the pentagon from the bathroom floor. Each corner of a regular pentagon is one hundred and eight degrees. Why can regular pentagons never cover a floor without a gap? Their corners never make a full turn. Three of them make three hundred and twenty-four, leaving a thirty-six degree gap, and four of them make four hundred and thirty-two, which overlaps. Being regular is not enough, then. A shape tessellates only if its corners meeting at a point make exactly three hundred and sixty. Now a different one: which of the five coaster shapes wastes the most cork? The circle. It has no straight edges and no corners, so circles packed side by side always leave gaps between them. One examiner's report notes this: Candidates sometimes produced an answer that showed no awareness of tessellation to maximise material use. In those answers, parts were fitted to the stock with no thought for which shapes pack together. The fix is to check the angles at a point before laying anything out, and choose squares, triangles or hexagons where the design allows. Those three shapes cut the most coasters from each cork sheet. Arranging real parts on a sheet is a skill of its own, and the video on nesting and efficient cutting takes it on.

Four questions to finish, from straight edges to octagons. Answer each one before the reveal.

Two angles sit side by side on a straight edge, and one is seventy degrees. The other? One hundred and ten degrees. Angles on a straight line add to one hundred and eighty, and one hundred and eighty take away seventy is one hundred and ten. Try this one: why is a regular hexagon's corner one hundred and twenty degrees? Two equilateral triangle corners meet there, and sixty plus sixty makes one hundred and twenty. Here's another wedge, opposite and adjacent both thirty millimetres. What is its angle? Forty-five degrees. Thirty over thirty is one, and tan to the minus one of one is forty-five. Equal short sides split the ninety evenly, which makes sense. One more: octagon corners are one hundred and thirty-five degrees. Do octagons tessellate? No, not on their own. Two corners make two hundred and seventy, and three make four hundred and five, so they never land on three hundred and sixty.

That is why the pentagon floor fails. One hundred and eight never builds up to three hundred and sixty, while the ninety of a square and the one hundred and twenty of a hexagon both do.

If you could answer all four of those, this one's done, so give it a thumbs up and it drops off your rewatch list. If one still feels unsure, keep the video saved, because angles settle with practice, and a few more questions usually does it.

Next in the chapter: Area and circumference: rectangles, triangles and circles, where a triangle's area turns out to be half a rectangle.

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