Test it: estimates, radii and angle recipes

In Discover and Understand you learned the moves: line up a protractor, swing a compass, bisect an angle. This layer is different. Here you are the scientist. Every chapter asks a what happens if…? or is it always true? question, asks you to commit to a prediction, and then tests it with a lab, a sheet of paper or a quick calculation.

You will find out whether long arms make a bigger angle, what a wrong-scale reading looks like, how good your eye really is, whether the compass radius matters, how small an angle you can reach by halving, which angles you can build from 60° and 90°, what your two set squares can make together, how accurate a pencil-and-protractor check can be, and why the bisectors you draw work every time.

Keep a geometry box, a few sheets of plain paper and a sharp pencil next to you. Many of the tests take less than a minute to do for real.

Draw two angles. The first has arms 3 cm long. The second has arms 12 cm long, four times as long. You open them by exactly the same amount. Which is the bigger angle?

Many learners, and quite a few adults, feel that the angle with long arms is "bigger". It looks bigger: it covers more paper. Before reading on, decide what you think.

There is a useful side to this fact. Because arm length does not change the angle, you are allowed to extend short arms before measuring. If an arm is too short to reach the scale of your protractor, lay a ruler along it and extend it lightly in pencil. You are drawing more of the same ray, so the angle you measure is the same angle.

Longer arms also make measurement more accurate, as you will discover in Chapter 8: a tiny wobble in where you draw the arm end matters much less when the arm is long.

A protractor carries two scales running in opposite directions. On one, 0 is at the right end of the base line; on the other, 0 is at the left end. At every mark, the two numbers add up to 180: where one scale says 40, the other says 140. (Which row is printed on the inside differs from protractor to protractor, so go by where the 0 is, not by “inner” or “outer”.)

The rule you learned is: line up one arm with the base line, then read the scale whose 0 sits on that arm. In this chapter you will investigate what happens when someone forgets the rule, and how to catch the mistake.

Look down the table and two patterns jump out.

1. Every wrong reading is 180 minus the right one. That is simply how the two scales are printed.
2. An acute angle always turns into an obtuse reading, and an obtuse angle into an acute one. Only a right angle survives the mistake: 90 on one scale is 90 on the other.

So the detective question is always the same: does my reading have the same type as the angle I can see? If the angle is clearly narrower than the corner of a page but you wrote a number bigger than 90, you read the wrong scale.

Estimating is not guessing. A good estimate uses benchmark angles you already know well: a right angle (the corner of a book), a straight angle (a ruler's edge), half a right angle (45°, a square folded corner to corner) and a third of a right angle (30°, the smallest corner of a 30-60-90 set square).

How close can you get just by looking? Before playing, predict your own accuracy.

Recall the 60° construction. Draw ray OA. With the compass point on O, draw an arc of any radius that cuts OA at P. Keeping the same radius, put the point on P and cut the first arc at Q. Draw ray OQ. Angle QOA is 60°.

Two things in that recipe are worth testing. First, does the size of the radius matter? Second, what happens if the radius accidentally changes between the two arcs?

The angle bisector splits any angle into two equal halves. If you can make 60°, one bisection gives 30°. Bisect again and you have 15°. Bisect once more: 7.5°. There is no limit to how many times you could bisect, at least on perfect paper with a perfect pencil.

With a ruler and compass you have a small set of moves: make 60° (an equilateral triangle), step round again to make 120°, make 90° (bisect between 60° and 120°, or draw a perpendicular), and bisect any angle you already have. You can also use a straight line, which gives 180°, and subtract from it.

The investigation: which angles can you reach by combining these moves? Before looking at the table, try to find routes to 45°, 75°, 105°, 135° and 150° on your own.

Notice the pattern in the right-hand column. Every recipe uses only three things: 60°, adding or subtracting angles that share an arm, and halving. Starting from 60° and 180°, halving gives 30°, 15°, 7.5°…; adding and subtracting mixes them. Everything you reach this way is a whole multiple of 15° or a halving of one.

So is every angle reachable? Try 20°, 40° or 10°. However you combine 60°, 90°, 180° and halving, you never land on them. The sort game below asks you to decide.

A geometry box has two set squares. One has angles 45°, 45°, 90°. The other has 30°, 60°, 90°. On their own they give you four angles: 30°, 45°, 60° and 90°. But you can place them side by side (adding their angles) or one on top of the other (subtracting). How many different angles can you make?

Collect the results and add the straight-angle trick (180° minus any of them), and the full list of angles from 15° to 180° you can draw with set squares is: 15°, 30°, 45°, 60°, 75°, 90°, 105°, 120°, 135°, 150°, 165°, 180°. That is every multiple of 15° up to 180°, and nothing else.

This is not a coincidence. Every set-square angle (30, 45, 60, 90) is a multiple of 15, and adding or subtracting multiples of 15 always gives another multiple of 15.

A construction is not finished until it is checked. After a 60° construction, lay the protractor on the vertex and read the angle. Will it show exactly 60? Almost never. It might show 59° or 61°. Is that a mistake?

To answer, find out how big one degree really is on paper. On a circle of radius 5 cm (a typical arc in a construction) a 1° slice has an arc length of only about 0.087 cm, a little less than 1 mm. A sharp pencil line is about half a millimetre wide. So the pencil itself can hide half a degree of error.

Two constructions have a special property that you can test with a divider (or a compass used as a measuring tool).

• Perpendicular bisector of AB: every point on it is equidistant (equally far) from A and from B.
• Angle bisector: every point on it is equally far from the two arms of the angle, where "distance to an arm" means the shortest distance, measured along a perpendicular.

Are these true only for the points where the arcs crossed, or for every single point on the line? Predict, then test.