Angles you can measure and make

Look at a patang (kite) before it goes up on Makar Sankranti. Two thin bamboo sticks cross each other, and the kite flies straight only if they cross at exactly the right angle. Look at a wooden photo frame: four pieces meet at the corners, and if even one cut is a little off, the corner shows an ugly gap. Look at a cricket pitch: the creases are painted at right angles to the pitch, or the umpire's decisions would be unfair.

In every one of these, someone had to measure an angle, or make one exactly. This lesson is about how people do that, with the small tin geometry box that sits in your school bag.

By the end you will be able to answer a surprising question: how do you draw an exact 60° angle with only a compass and a ruler, and no protractor at all?

An angle is made when two straight lines, called arms, start from the same point, called the vertex. The size of the angle tells you how much you have to turn to go from one arm to the other.

You already know some angles by sight. The corner of your notebook is a right angle. A door opened just a little makes a small angle with the wall; a door flung wide open makes a big one. But "a little" and "wide" are not good enough for a carpenter, a builder or a kite maker. They need to say exactly how big an angle is, and then draw or cut it exactly. For that we need a unit, the degree, and tools that work with degrees.

Almost every Indian school child owns a small metal or plastic geometry box (some people call it an instrument box). Open it and you will usually find the same set of tools. Each one has a job. Using the right tool for the job, and using it carefully, is half the secret of good geometry.

Good geometry is careful geometry. A few habits make a huge difference:

• Sharpen your pencil to a fine point. A thick line can be half a millimetre wide, and then nobody can tell exactly where it is.
• Tighten the compass. If the hinge is loose, the legs slide apart while you draw, and your circle will not close.
• Look at the ruler's zero. On many rulers, 0 is not at the very end. Measure from the 0 mark, not from the edge.
• Keep the protractor clean and flat. A scratched or bent protractor gives wrong readings.
• Draw lightly first. Construction arcs are helpers, so draw them faintly and keep them. They show your working.

Stand up and turn right round until you face the same way again. That is one full turn. Mathematicians split a full turn into 360 equal tiny turns, and each tiny turn is called one degree, written 1°.

• A full turn is 360°.
• A half turn (facing the opposite way) is 180°. This is a straight angle: the two arms make a straight line.
• A quarter turn is 90°, a right angle, like the corner of a page.

One degree is really small. If you turned only 1° you would hardly notice you had moved.

Why 360 and not 100? Nobody is completely sure, but the idea probably comes from the ancient Babylonians of Mesopotamia, more than 4,000 years ago. They counted in groups of 60, and a year is close to 360 days, so the Sun seems to move about one degree across the sky each day.

There is also a very practical reason 360 has lasted: it splits evenly in a huge number of ways. 360 has 24 whole-number divisors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360. So a half, a third, a quarter, a fifth, a sixth, an eighth, a ninth, a tenth and a twelfth of a turn are all whole numbers of degrees. Compare 100, which has only 9 divisors, or 365, which has only 4 (1, 5, 73 and 365).

Just how small is one degree? Imagine two long straight sticks, each 1 metre long, joined at one end. Open them to 1° and the far ends are only about 1.7 cm apart, roughly the width of your thumb. That is why a sharp pencil and a careful eye matter: a sloppy line can easily be a degree or two off. For school work, being within 1° or 2° of the true size is counted as accurate.

Before a good carpenter picks up a measuring tool, they look and guess. Guessing first is not cheating. It is a safety net. If your guess says "about 40°" and your measurement says 140°, you know at once that something went wrong.

Here is a simple way to estimate any angle:

1. Compare it with a right angle. Is it smaller than the corner of a page (acute), bigger (obtuse), or exactly the same?
2. Compare with half a right angle (45°). If it is acute, is it thinner or fatter than half a right angle?
3. Use the clock. Each hour gap is 30°. Picture clock hands to see 30°, 60°, 90°, 120° and 150°.
4. Say a number. Commit to a guess, like "about 70°".

Take the protractor out of your box and look closely. It has three parts that matter:

• The centre point: a tiny hole, cross or dot in the middle of the straight edge. This is where the vertex of your angle must go.
• The base line: the line on the protractor that runs through the centre point from 0 on one side to 180 on the other. One arm of your angle must lie along it.
• Two scales: two rows of numbers from 0 to 180 running round the curved edge. One row counts up from the right, the other counts up from the left. Which of the two rows is printed on the inside and which on the outside is not the same on every protractor, so never go by “inner” or “outer”. Go by where the 0 is.

Why two scales? So you can measure an angle that opens either way, without turning the protractor upside down.

Here is the one rule you need for the two scales:

Use the scale that shows 0 on the arm you lined up with the base line.

Follow that scale round, counting up from 0, until you reach the other arm. The number where the other arm crosses is the size of the angle. The other scale at the same spot will show a different number, and the two numbers always add up to 180. That is why a quick estimate is so useful: it tells you which of the two numbers makes sense.

Measuring is reading an angle that is already there. Drawing is the opposite: you start with a number, like 50°, and make an angle of exactly that size. The protractor does both jobs.

Now for the puzzle from Chapter 1. Can you make an exact 60° angle with only a compass and a ruler, and no numbers at all?

Yes, and it is one of the oldest tricks in mathematics. The ruler is only used to draw straight lines, never to measure. Drawing like this, with only a straight edge and a compass, is called a construction.

Once you can make 60°, a whole family of exact angles opens up. Step round the arc twice from P and you get 120°. Cut an angle exactly in half, which is called bisecting it, and 60° becomes 30°. A right angle, 90°, can be made in more than one way with a compass, and cutting it in half gives 45°. You will learn each of these, and why they work, in the next layers.

Your geometry box also holds two triangles of plastic or metal, called set squares. Each one carries a few angles you can trace straight away, with no measuring and no arcs:

• The 45° set square has angles of 45°, 45° and 90°.
• The 30°–60° set square has angles of 30°, 60° and 90°.

So with set squares alone you can draw 30°, 45°, 60° and 90° angles in a second. Engineers and draughtspeople used them all day long before computers.