Reading and drawing an angle with a protractor

0:00Last time you folded your own protractor out of paper, and got as far as creases every twenty-two and a half degrees. Now let's take the manufactured one, out of a geometry box, and look at what it actually is. Start with a version your book shows first: the same half-disc, with all its tick marks, but no numbers printed on it at all. That straight edge along the bottom is a straight angle. A hundred and eighty degrees.

0:31And it's been divided into a hundred and eighty equal parts. One tick, one degree. Look closely at the ticks, because they're not all the same length, and that's deliberate. Every tenth mark is long. Halfway between two long ones, there's a medium one — that's the five. And all the little short ones in between are the single degrees. So even with no numbers on it, that thing is completely usable. The information is in the marks, not the printing.

1:05Your book proves that by making you read one the hard way. There's a figure with a vertex at A, sitting on the centre of the unlabelled protractor, and rays going out to points K and L. You want angle K A L. So find where the arm A K crosses the scale, and find where A L crosses it. And then count the ticks between them. One at a time.

1:31One, two, three, four, five… and you keep going, and you get to thirty. Thirty unit parts. So the angle is thirty degrees. That's it. No numbers were needed, because the count was the measurement. This is the same idea as the last video, and I want it to feel almost boringly familiar by now. Measuring is counting unit parts. But you probably noticed something while I was counting. It was awful. Nobody wants to count to a hundred and thirty-five one tick at a time.

2:08So use the marks. That's what they're for. Jump long mark to long mark and you're going up in tens. Ten, twenty, thirty, forty. Land on a medium mark and you've got a five to add. Then only the last few short ticks get counted one by one. Same answer, a fraction of the effort. Tens, then fives, then ones. And now you can see why somebody eventually printed the numbers on. It's not a different method. It's the counting, done in advance, at the factory.

2:45So here's the labelled one. The one you've actually got. The numbers run round the rim, and each one is just telling you the count from one end. Which saves you the counting entirely — you read where the arm crosses, and the number's already there. But now something appears that confuses almost everybody the first time. There isn't one row of numbers. There are two. One row runs zero, ten, twenty, thirty, up to a hundred and eighty, going one way round.

3:18The other row runs the same values going the other way. So at every mark there are two numbers, and they always add to a hundred and eighty. Your book calls them the inner and outer markings. And the question people ask is: which is the right one? It's the wrong question, and I want to fix it now rather than later. Neither row is the right one. Neither row is the wrong one either.

3:47They're there so that you can put your zero at whichever end suits you. If the angle you're measuring opens to the right, you want a row whose zero is on the right-hand side. If it opens the other way, you want the other row. So the rule isn't use the outer scale, or use the inner scale. The rule is: use the row whose zero sits on the arm you started from.

4:13That's the entire purpose of printing two. One tool, usable from either direction, without turning it over. Now here's the thing that makes all of this make sense, and your book does it properly instead of hiding it. Take a protractor with its centre at a point O, and six rays leaving O — call them O P, O Q, O R, O S, O T and O U. Read where each one crosses the scale, going from the P end.

4:45O P is at zero. O Q is at thirty-five. O R is at ninety-five. O S is at a hundred and twenty-five. O T is at a hundred and sixty. And O U is at a hundred and eighty. Six numbers. And now every single angle in that figure is a subtraction between two of them. Angle Q O R? Ninety-five take away thirty-five. Sixty degrees. Angle R O S? A hundred and twenty-five take away ninety-five. Thirty degrees.

5:21Angle S O T? A hundred and sixty take away a hundred and twenty-five. Thirty-five degrees. Angle Q O T? A hundred and sixty take away thirty-five. A hundred and twenty-five degrees. You can do every pair in that figure, and there are fifteen of them, without ever moving the protractor. So a protractor doesn't give you an angle. It gives you two readings. You do the subtracting. And here's the proof that the two rows really are equivalent.

5:55Read angle T O S again on the other row. Now O T is at twenty and O S is at fifty-five. Different numbers. Fifty-five take away twenty is thirty-five. The same answer. The scale you choose changes the two numbers, and it never changes the difference. Right. Now the shortcut, and it's a shortcut, not a rule handed down from nowhere. Subtracting is fine, but subtracting from zero is easier, because there's nothing to subtract.

6:30So slide the protractor until one of the arms is lying exactly along the zero mark. Now that arm reads zero. And the other arm's reading, whatever it is, is the answer, because you'd be taking zero away from it. That is the whole reason for the instructions you've been given. Centre on the vertex, so both arms cross the scale from the same point. One arm on the zero, so that one of the two readings disappears.

7:02It was never a ritual. It's an arrangement that makes the arithmetic vanish. And that's worth knowing, because if you ever can't line an arm up with zero — the angle's in the middle of a drawing, or the paper's in the way — you're not stuck. You just do the subtraction, like we did a minute ago. Your book then has a page I really like, called Mind the Mistake, Mend the Mistake.

7:30Six panels. In each one, somebody has put a protractor on an angle and written down a number. Thirty-five degrees. Eighty. Seventy. A hundred and fifty. A hundred and twenty. Eighty-five. And every single one of those six numbers is wrong. Not because the arithmetic slipped. Because the protractor was in the wrong place. In one, the vertex isn't on the centre — it's drifted off to the side. In another, the arm isn't on the zero line. It's been lined up with the bottom edge of the plastic instead.

8:09And those are not the same thing. On most protractors the zero line is printed a little way above the physical edge. If you use the edge, you're wrong before you start. In others the protractor's tilted, or flipped over so the rows have swapped. Now, here's what all six have in common, and it's the real lesson. In every case the instrument gave a number. A confident, specific, wrong number.

8:39A protractor never tells you that it's badly placed. It just reports the gap between two crossings, and if the crossings are wrong, so is the gap. So the checking has to be yours. Vertex on centre. Arm on the zero line, not the edge. Read the row whose zero is on that arm. Six mistakes, one idea: the placement is the measurement. So far we've been reading. Now let's draw, and you'll notice it's the same procedure backwards.

9:14Your book asks for an angle of thirty degrees, called angle T I N. Step one. Draw a ray. Start at the point I and draw out to N. That's your first arm, and it's going to be the zero arm. Step two. Put the protractor's centre on I, and lay I N exactly along the zero line. Step three. Now find thirty on the row whose zero is on I N, and make a small mark on the paper there. Call it T.

9:45Step four. Take the protractor away, and join I to T with a ruler. And there it is. Angle T I N, thirty degrees. Read the four steps again and compare them with measuring: same placement, same row, same number — you're just supplying the number and finding the arm, instead of finding the arm and reading the number. Now a good problem. What if the angle is bigger than the protractor?

10:16Your book has one where the opening you're asked about is huge — much more than a hundred and eighty degrees — and your protractor only goes up to a hundred and eighty. It looks like a dead end. It isn't. Because the two arms make two angles, remember — the one you're being asked about, and the rest of the way round. And together, those two are a full turn. Three hundred and sixty degrees.

10:45So measure the other one. The small one. The one that does fit on your protractor. In your book's figure that comes out at a hundred degrees. Now subtract: three hundred and sixty take away a hundred. Two hundred and sixty degrees. Done. And notice you measured the angle you weren't asked about, on purpose, because it was the measurable one. That's a habit worth having well beyond this chapter — when the thing in front of you won't measure, measure its companion and subtract.

11:20Your book finishes with two games, and they're better than they look. In the first, somebody draws an angle and everyone guesses its size before it's measured. In the second, somebody calls out a size and everyone draws it by eye, and then you measure what they drew. And here's the scoring, which is the clever part. Your score is how far off you were. Guess thirty-nine when it was really forty-nine? That's ten. Draw twenty-five when thirty-four was called? That's nine.

11:55And the lowest total wins. So being over by ten is exactly as bad as being under by ten. There's no way to game it by always guessing high or low. And the point isn't the game. It's that estimating is a real skill, and it's separate from measuring. There's a note in your book asking teachers to play it again on different days, which tells you it's meant to improve with practice, like anything else.

12:25So try it now, before we finish. Look at the angle on screen. Say a number out loud. Then we'll measure it, and you'll find out how far off you were. One last idea, and it's the one that takes angles off the page. Your book asks where the angles are in ordinary things, and the interesting part is that usually one arm isn't drawn anywhere. A clock. The two hands are the arms, the pin at the middle is the vertex. That one's easy — both arms are visible.

13:00And the face is already cut into twelve equal parts, so each hour interval is three hundred and sixty divided by twelve. Thirty degrees. So at one o'clock the hands are thirty degrees apart. At two, sixty. At three, ninety — a right angle, which you knew. At four, a hundred and twenty. At six, a hundred and eighty, and they're in a straight line. But be careful past six, because the pattern stops answering the question.

13:33At nine o'clock the hour hand has turned two hundred and seventy degrees from twelve — but the two hands are only ninety degrees apart. Three hundred and sixty take away two hundred and seventy. So after six, keep subtracting from three sixty, or you'll be quoting the reflex angle instead of the gap. Then the harder ones. A door standing open — one arm is the door, and the other arm is the wall, which nobody drew.

14:06A ramp — one arm is the slope, the other is the flat ground it rises from. A swing at the top of its arc — one arm is the rope now, the other is where the rope hangs when it's still. In each of those, half the work is deciding what the second arm is. Once you've drawn it in, it's just a protractor reading. Now, here's yours, and it's a real one from your book.

14:32Draw three triangles, all different shapes. Measure all three angles of each one, and add them up. You'll get the same total every time, and I'm not going to tell you what it is or why. Your book won't either — it says outright that the reason comes in a later year. For now it's a conjecture you found by measuring, which is a perfectly respectable thing for it to be.

14:59Measure carefully, and tell me your three totals. Comments below. Next time: sorting every angle there is into acute, obtuse and reflex — including the big one we've been ignoring since the very first video.