A protractor has the answer printed on it. A compass does not. When you construct an angle with a compass and a ruler, nothing is measured and nothing is read off a scale. The angle appears because of what a circle is. That is why these are called constructions and not drawings, and it is why your teacher will ask you to put the protractor away until the very end, when you use it only to check.
The one move everything is built from
Open your compass to some width. Do not change it again. Draw a circle. Now put the compass point anywhere on the circle you just drew and make a mark on the circle. Move the point to that mark and mark again. Keep going. You will come back to exactly where you started after six steps, never five, never seven, and it does not matter how wide you opened the compass.
Six equal steps make one full turn. A full turn is 360 degrees. So each step must be 360 divided by 6, which is 60 degrees. Nobody measured anything. The 60 came out of the circle by itself.
Look at what you drew. The centre of the circle, the point where you started, and the mark you made are three corners of a triangle. Two of its sides are radii of the circle, so they are equal. The third side is the compass width, and you set the compass to the radius, so that side is equal too. All three sides are the same length. A triangle with three equal sides has three equal angles, and the three angles of any triangle add up to 180. So each angle is 60. That is the whole proof, and it fits in four sentences.
The second move: cutting an angle in half
The compass can do one other thing. Given any angle at all, it can cut that angle into two equal halves without measuring it. This is called bisecting. Put the compass point on the corner and draw an arc across both arms. Now put the point on each of those two crossing places in turn, and with the same compass width draw two arcs that cross each other out in the middle. Join the corner to that crossing point. That line splits the angle exactly in two.
Every angle in this lesson is made from just two moves: step the radius round a circle to get 60, and cut an angle in half. Adding 60s gives you 60, 120 and 180. Halving gives you 30, 15 and 45. Adding those together gives you everything else. There is no third trick, and once you see this you never have to memorise nine separate recipes again.
The family tree of every angle you need
| Angle | How it is made | Move used |
|---|---|---|
| 60° | One radius step round the circle | Step |
| 120° | Two radius steps, 60 + 60 | Step |
| 180° | Three radius steps, a straight line | Step |
| 30° | Half of 60 | Halve |
| 90° | Halfway between 60 and 120, which is 60 + 30 | Halve |
| 45° | Half of 90 | Halve |
| 15° | Half of 30 | Halve |
| 75° | 60 + 15, or equally the halfway point between 60 and 90 | Both |
| 105° | 90 + 15, the halfway point between 90 and 120 | Both |
| 135° | 90 + 45, the halfway point between 90 and 180 | Both |
| 150° | 120 + 30, the halfway point between 120 and 180 | Both |
Read the last four rows again. 75 sits halfway between 60 and 90. 105 sits halfway between 90 and 120. 135 sits halfway between 90 and 180. 150 sits halfway between 120 and 180. Every one of the awkward angles is simply the bisector of two angles you already have. So the real question is never how do I make 105. It is which two angles is 105 sitting between.
Copying an angle you cannot measure
There is one more thing the compass does, and it is the one that matters most outside the classroom. Suppose somebody hands you an angle that is not a nice number at all, something like 37.4 degrees. You can copy it exactly onto a new line without ever finding out what it measures. Draw an arc across both arms of the original. Draw the same arc on your new line. Then set your compass to the gap between where the arc crosses the two arms, and step that gap across on your new arc. Join up. The two angles are now identical, and you still have no idea what either of them measures.
A carpenter cutting rafters for a roof does not measure the angle at the ridge and then measure it again on each rafter. Errors would pile up. Instead the first rafter becomes the pattern and every other one is copied from it, which is exactly the compass method in wood. The same reasoning is why a tailor keeps a paper pattern and why a mason uses a set square rather than re-measuring every corner. Copying is more accurate than measuring, because measuring introduces a fresh error every single time.
Changing the compass width partway through. The moment you nudge that screw, the triangle stops being equilateral and your 60 quietly becomes 58 or 63. Nothing on the page will warn you. Set the width, and then hold the compass by the very top and turn it, never by the legs, because holding the legs is what squeezes them. If your check with the protractor comes out two or three degrees off, this is almost always the reason.
You can halve any angle with a compass. You cannot cut every angle into three. Ask for 20 degrees, which is a third of 60, and no compass and ruler in the world will give it to you exactly. People tried for two thousand years before it was finally proved impossible. So the angles in your table are not a random list somebody chose. They are very close to the complete list of what this tool can reach, and 20 is missing for a real mathematical reason, not because the book forgot it.
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Every construction below is written the way you should write it in an exam: what you are given, what you want, then numbered steps, then the check. Marks are given for the steps, not just the final picture.
Construct an angle of 90° using only a compass and a ruler
| Step | What you do | Why |
|---|---|---|
| 1 | Draw a ray OA. | This is the fixed arm. |
| 2 | With O as centre and any radius, draw an arc cutting OA at P. | This radius will now be used for everything. |
| 3 | With P as centre and the same radius, cut the arc at Q. | Angle QOP is 60°. |
| 4 | With Q as centre and the same radius, cut the arc at R. | Angle ROP is 120°. |
| 5 | With Q and R as centres and a radius more than half of QR, draw two arcs meeting at S. | This bisects the gap between 60 and 120. |
| 6 | Join OS. | Angle SOP = 60 + 30 = 90°. |
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With the compass set to the radius, stepping round a circle gives 60° every time. Six steps close the circle.
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