What this lesson is really about

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.

Why exactly six, and never five or seven

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.

The whole lesson in one line

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

A pattern worth noticing

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.

Where this is actually used

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.

The mistake almost everyone makes

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.

Something the compass genuinely cannot do

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.

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.

Worked Example 1

Construct an angle of 90° using only a compass and a ruler

Think firstWhere does 90 sit? Radius steps give me 60 and 120. And 90 is exactly halfway between 60 and 120. So this is a bisection problem, not a new construction.

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°.

CheckMeasure angle SOP with a protractor. It should read 90°. If it reads 88 or 92, your compass width slipped between steps 3 and 4.

Worked Example 2

Construct an angle of 105°

Think first105 is not on the free list. But 90 and 120 are, and 105 is exactly halfway between them, because 90 + 120 = 210 and half of 210 is 105. So build 90, build 120, then bisect the gap.

Step	What you do	Running total
1	Draw ray OA. Arc from O cuts it at P.	0°
2	Step the radius from P to Q.	60°
3	Step it again from Q to R.	120°
4	Bisect angle QOR. Call the new ray OS.	90°
5	Bisect angle SOR, the gap between the 90 ray and the 120 ray.	105°

Notice how short this gotOnce you have the 90 ray and the 120 ray on the page, 105 costs you one extra bisection. And 75 costs you one bisection on the other side, between 60 and 90. Draw all your rays first, then harvest the angles you need.

Worked Example 3

Copy a given angle onto a new ray, without measuring it

GivenSome angle ABC. You are not told what it measures and you must not measure it.

Step	What you do
1	With B as centre and any radius, draw an arc cutting BA at P and BC at Q.
2	Draw a new ray YZ. With Y as centre and the same radius, draw an arc cutting YZ at M.
3	Set the compass to the distance PQ. Do not change it.
4	With M as centre and that width, cut the new arc at N.
5	Join YN. Angle NYZ is equal to angle ABC.

Why this worksTriangle BPQ and triangle YMN have three matching sides: the two radii are equal because you kept the same radius, and PQ equals MN because you copied that distance. Three equal sides force the triangles to be identical, so the angle at B must equal the angle at Y. This is the same equal-sides argument that gave you 60 in the first place.

Group work

Six angles, six groups

MethodDivide into groups. Each group constructs one of 60°, 120°, 75°, 135°, 150° and 90°, then presents it to the class. Show your arcs. Do not rub them out, because the arcs are the proof that you constructed the angle rather than drew it.

Ask each group one questionWhich two angles did yours sit between, and how many times did you have to bisect? The 60 and 120 groups will answer none. Everyone else will answer once or twice. That answer is the real content of the lesson.

Evaluation

Change the compass width and prove it does not matter

MethodConstruct 60°, 120°, 75°, 135°, 150° and 90° again, but this time every student uses a different arc size. Small compass, wide compass, whatever you like. Then measure all of them with a protractor and compare across the class.

What you are provingForty different arc sizes, forty identical answers. The width you choose changes the size of the picture and changes nothing about the angle. That is the difference between a construction and a drawing, and this activity is the only place in the lesson where you actually see it.

Try This Too

Fold a piece of paper instead

MethodTake a rectangular sheet. Its corner is already 90°. Fold that corner so one edge lands exactly on the other edge. Open it out. The crease is at 45°. Fold again the same way and you have 22.5°. No compass, no protractor, and the same halving move.

Then think about thisFolding can halve an angle just as well as a compass can. So try to fold 20°, a third of 60. You will not manage it, and neither will anybody else, ever. Halving is easy for both tools. Cutting into three is beyond both of them.

Worked Example

One group was given 135°. Instead of hunting for a recipe, they wrote down what they already had and worked backwards. Here is the page they handed in.

Reflection

What surprised us was that we never needed to look up how to make 135. We only needed to know two things: which angles come free, and that bisecting exists. Everything else was working backwards from 135 until we hit something free. Our teacher said this is what mathematicians actually do, and that the recipes in the back of the book are just somebody else's working written out neatly.

Your TurnDo the same for an angle your teacher gives you. Fill the table in before you touch the compass. If you cannot fill row three, you are not ready to start drawing yet.

Question	Your answer
My angle is	-
Is it free from radius steps alone?	-
Which two angles does it sit between?	-
How many bisections in total?	-
Protractor check	-

Care with the compass

The point is genuinely sharp. Keep it pointing down at the paper, never towards yourself or anybody beside you, and put the compass flat on the desk when you are not using it rather than leaving it standing up. Press hard enough that the point does not skid, but not so hard that it goes through the page, because a torn hole makes the next arc impossible to centre.

With the compass set to the radius, stepping round a circle gives 60° every time. Six steps close the circle.

60° comes free because the centre, the start point and the mark form an equilateral triangle, and 180 ÷ 3 = 60.

The width you open the compass to never changes the angle. It only changes the size of the picture.

Radius steps alone give 60°, 120° and 180°. Nothing else is free.

Bisecting cuts any angle into two equal halves without measuring it.

30° is half of 60°. 15° is half of 30°. 45° is half of 90°.

90° is the bisector of 60° and 120°.

75° is the bisector of 60° and 90°. 105° is the bisector of 90° and 120°.

135° is the bisector of 90° and 180°. 150° is the bisector of 120° and 180°.

An angle can be copied exactly onto a new ray without ever measuring it, using one arc and one distance.

Never change the compass width in the middle of a construction. This causes almost every wrong answer.

Leave your arcs on the page. They are the working, and marks are given for them.

The protractor is for checking at the end, never for building.

20° cannot be constructed with compass and ruler at all. Halving is possible, cutting into three is not.

1Why does stepping the compass round a circle always give 60°, whatever width you set?

Because the compass is set to the radius. The centre of the circle, the point you start from and the mark you make are joined by three lines: two of them are radii of the circle, and the third is the compass width, which you set equal to the radius. So all three sides of that triangle are the same length, which makes it equilateral. The three angles of any triangle add up to 180°, and in an equilateral triangle all three are equal, so each one is 180 ÷ 3 = 60°. The size of the radius never enters the argument, which is exactly why the width does not matter.

2Write the steps to construct an angle of 120° with a compass.

Draw a ray OA. With O as centre and any radius, draw an arc cutting OA at P. Keeping exactly the same radius, put the compass point on P and cut the arc at Q. Keeping the same radius still, put the point on Q and cut the arc again at R. Join OR. Angle ROA is 120°, because you have taken two radius steps round the circle and each step is 60°.

3Explain how you would construct 45°, and say how many bisections it needs.

It needs two bisections. First make 90°, which is itself a bisection: step the radius twice to get the 60° ray and the 120° ray, then bisect the gap between them, giving 90°. Then bisect the angle between the 90° ray and the original ray, which splits 90 into two halves of 45 each. So the chain is 60 and 120 free, one bisection for 90, a second bisection for 45.

4A student constructs 75° and the protractor reads 78°. Give two likely reasons.

First, the compass width slipped between arcs. If the two arcs that are supposed to be equal are not equal, the triangle is no longer equilateral and the 60° that everything is built on is already wrong before the bisection even starts. Second, the bisecting arcs were drawn with too small a radius, so they crossed at a very shallow angle and the exact crossing point was hard to see, which throws the final ray off. A third and simpler possibility is that the protractor was read from the wrong scale, since most protractors carry two rows of numbers running in opposite directions.

5How can you copy an angle onto another ray without measuring it, and why does the method work?

Draw an arc from the corner of the original angle so that it crosses both arms, at P and Q. Draw a new ray and, with the very same radius, draw an arc from its end point, crossing it at M. Now set the compass to the straight distance from P to Q and, with M as centre, cut the new arc at N. Join the end point to N. The method works because the two triangles formed have three pairs of equal sides: two pairs are equal because the same radius was used both times, and the third pair is equal because the distance PQ was copied directly. Three equal sides force the two triangles to be identical in every way, so the angles at their corners must be equal too.

6Which two angles does 135° sit between, and why is that the fastest way to think about it?

It sits exactly between 90° and 180°, because 90 + 180 = 270 and half of 270 is 135. Thinking this way is fastest because it turns a construction you have never done into one you have: you already know how to make 90 and you already know that 180 is just a straight line, so the only new work is a single bisection. The alternative, trying to remember a separate recipe for 135, gives you nothing you can reuse when the exam asks for 105 instead.

7Why should you not rub out your arcs before handing in the work?

Because the arcs are the evidence that the angle was constructed rather than drawn with a protractor. A finished ray on its own proves nothing, since anybody could have measured it. The arcs show which centres you used, that the radius stayed the same, and which gap you bisected, and marks in a construction question are awarded for exactly those things. Rubbing them out throws away most of the marks and leaves a picture that could have been produced any way at all.

8Can 20° be constructed with a compass and ruler? Explain.

No. 20° is one third of 60°, and cutting an angle into three equal parts cannot be done with only a compass and a straight edge. Halving is always possible, which is why 30° and 15° are easy, but trisecting is not, and this was finally proved impossible in the nineteenth century after people had attempted it for around two thousand years. So the list of angles in this lesson is limited by the tool itself and not by the syllabus: every angle you are asked for is built from 60° by adding and halving, and 20° cannot be reached that way.

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The video for this topic is being prepared. Check back soon!

Unit 1 · Lines and Angles

Constructing Angles with a Compass

Mathematics · Grade 7

What You Will Be Able To Do

📐

Construct 60, 120, 90, 30, 45, 75, 105, 135 and 150 degrees with compass and ruler only

✂️

Cut any angle exactly in half without measuring it

📋

Copy an angle onto a new ray when you are not told what it measures

🧠

Explain why the answer is 60 and why the compass width makes no difference

Walk the Radius Round the Circle

1Open the compass to any width. Draw a circle. Do not touch the screw again

2Put the point anywhere on the circle and mark the circle

3Move to that mark. Mark again. Keep going all the way round

4You land back at the start after exactly six steps. Never five, never seven

5Six equal steps make 360, so one step is 360 divided by 6, which is 60

“

Two sides are radii of the circle. The third side is the compass width, and the compass was set to the radius. So all three sides are equal, the triangle is equilateral, and 180 shared between three equal angles is 60.

The Only Two Moves There Are

👣 Step

Walk the radius round the circle

Gives 60, then 120, then 180

Costs nothing. These three are free

✂️ Halve

Bisect any angle at all, without measuring

Gives 30 from 60, 15 from 30, 45 from 90

Everything not on the free list comes from here

Add and halve. That is the whole toolkit

What You Get For Free

1️⃣

60 degrees. One radius step. The equilateral triangle does the work

2️⃣

120 degrees. Two radius steps. Not a new construction, the same one twice

3️⃣

180 degrees. Three radius steps, which is simply a straight line

🚫

Nothing else. Every other angle has to be paid for with a bisection

Building 105 in Five Moves

1Ray OA. Arc from O cuts it at P. Running total 0

2Step the radius from P to Q. Running total 60

3Step it again from Q to R. Running total 120

4Bisect angle QOR to get the 90 ray. Running total 90

5Bisect the gap between the 90 ray and the 120 ray. Answer 105

GROUP WORK · 25 MIN

Forty Different Compasses, One Answer

BUILD · 15 MIN

Six groups, one angle each: 60, 120, 75, 135, 150 and 90. Every student uses a different arc size

MEASURE · 5 MIN

Now the protractors come out. Every student checks their own and writes the reading on the board

COMPARE · 5 MIN

Forty arc sizes, forty identical readings. The width changed the picture and changed nothing else

Presenter notes: Open by holding up a protractor and a compass, one in each hand. Ask the class which one is cheating. Let them argue for a moment, then say it plainly: the protractor already has every answer printed on it, so using it to make an angle is copying. The compass has no numbers anywhere on it, and yet it can produce a perfect sixty degrees. Today we find out how something with no numbers on it can know what sixty degrees is. Tell them the protractor stays in the box until the last five minutes of the lesson, when it is used only to check. Two periods are allotted to this topic.