What the two tools may do
A construction uses two tools and nothing else. The straightedge draws a straight line through two points that are already on the page, and it has no marks to measure with. The compass draws a circle, or part of one, centered on a point already on the page, with a width already on the page. New points appear only where these lines and circles cross.
Every construction so far keeps to these rules: the perpendicular bisector, the angle bisector, the perpendicular from a point, and the triangle from three sides.
Halving is always possible
To bisect an angle, draw one arc from the vertex across both arms, draw equal arcs from the two cuts, and rule from the vertex through their crossing. The method never asks how big the angle is, so it halves every angle, exactly. Copying an angle is possible too: one arc fixes how far apart the arms are, and the compass carries that width to a new place.
A 60° angle bisected with a compass and straightedge into two angles of 30°.
The question about thirds
To trisect an angle is to cut it into three equal angles. The question is whether there is one construction, with these two tools only and a finite number of steps, that trisects every angle, as the bisector construction halves every angle.
The question was asked by geometers in ancient Greece. For more than two thousand years people looked for such a construction, and no one found one.
Some angles can be trisected
A right angle can. Draw an arc from the vertex V that cuts one arm at X. Keep the same width and draw an arc from X that crosses the first arc at E. Then VX, VE and XE are all one compass width, so VXE is an equilateral triangle. It has three lines of symmetry, so its three angles are equal, and since they add up to 180°, each is 180° ÷ 3 = 60°. So ∠XVE = 60°, and the rest of the right angle is 90° − 60° = 30°. Bisect the 60° and the right angle is cut into 30°, 30° and 30°.
A 60° angle cannot be trisected. Trisecting it would construct an angle of 20°, and an angle of 20° cannot be constructed with these tools.
So the impossibility is about a single method for every angle. A construction that works for one special angle is not a trisection of angles in general.
An angle of 20°, a third of 60°. No compass-and-straightedge construction produces it.
A method that looks right and is not
Here is a tempting attempt. Draw an arc from the vertex that cuts both arms, join the two cuts with a straight line called a chord, and cut the chord into three equal parts. Cutting a segment into three equal parts is possible with a compass and straightedge. Then rule from the vertex through the two points that divide the chord.
It does not trisect the angle. The middle of the chord is closer to the vertex than its two ends, so the middle third of the chord takes up a wider angle than the outer thirds. On a right angle, each outer angle measures about 26.6° and the middle one about 36.9°, where a trisection would give 30° each.
The two axes make a right angle, and an arc of radius 3 from the origin cuts them at (3, 0) and (0, 3). The chord between those points is cut into three equal parts at (2, 1) and (1, 2), and the rays through those points make angles of about 26.6°, then 36.9°, then 26.6° again: the middle angle is the largest.
Proved impossible in 1837
In 1837 the French mathematician Pierre Wantzel proved that no construction can trisect a 60° angle. The idea of the proof is about which lengths the tools can reach, starting from a length of 1.
Every new point in a construction is a crossing of lines and circles. The equation of a straight line has no squared terms, and the equation of a circle, such as , has squares but nothing higher. So finding a crossing means solving an equation with a square at most, and the answer uses only adding, subtracting, multiplying, dividing and square roots. Every length a construction can reach is built from whole numbers in those ways, and in no other way.
Trisecting 60° would reach a particular length x. Draw a circle of radius 1 around the vertex, mark where the 20° arm cuts it, and drop the perpendicular from that mark to the first arm: x is the distance from the vertex to the foot of that perpendicular. Trigonometry, a later topic, shows that this length solves , an equation with a cube in it. Wantzel showed that its solution cannot be written with square roots alone. So no construction reaches that length, and no construction trisects 60°. One angle that cannot be trisected is enough to show that there is no method that trisects every angle.
What impossible means here
It does not mean that nobody has found the method yet. It is proved that none exists, so every claimed trisection with a compass and unmarked straightedge alone has a mistake in it somewhere, like the chord method above.
It is about these two tools only. With a ruler carrying two marks, Archimedes trisected any angle, and folding paper can trisect an angle too.
And it is about an exact result in a finite number of steps. You can get as close to a third as you like, because and each of those fractions of an angle is found by bisecting again and again. A quarter of the angle, plus a sixteenth, plus a sixty-fourth, and so on, comes nearer and nearer to a third, but after any finite number of steps it is still short of a third.
The same 1837 proof showed that doubling a cube, constructing the edge of a cube with twice the volume of a given one, is impossible too, because it needs the cube root of 2.