Triangles, polygons and the impossible angle

You can now measure any angle, draw any angle, and construct 60°, 90°, 30°, 45° and their relatives with a compass and straightedge. You also know why those constructions work. This last layer asks: what can you build with them, and where do they run out?

You will construct triangles from just three measurements, draw regular polygons inside a circle, meet a 17-sided shape that a teenager in Germany proved could be constructed, and learn why an innocent-looking angle like 20° can never be constructed exactly with ruler and compass, no matter how clever you are. Along the way there are projects to make with your hands, olympiad-style puzzles, and a look at the people who use angles for a living.

How much do you need to know about a triangle to draw it exactly? A triangle has six measurements: three sides and three angles. Surprisingly, three well-chosen facts are enough to fix the whole triangle. Anyone, anywhere, following the same three facts will draw a copy of exactly the same size and shape.

The four standard sets of facts are:

• SSS: all three sides.
• SAS: two sides and the angle between them (the included angle).
• ASA: two angles and the side between them.
• RHS: a right angle, the hypotenuse (the side opposite the right angle) and one other side.

These are exactly the conditions that make two triangles congruent (identical in shape and size), which is the reason the constructions in Deepen work.

A regular polygon has all sides equal and all angles equal. The neat way to draw one is to start with a circle and split the full turn at its centre into equal central angles: a regular n-sided polygon has central angle 360° ÷ n. Mark those points on the circle and join them.

So drawing a regular polygon with ruler and compass is really the question: can I construct the central angle 360° ÷ n?

Which regular polygons can be constructed with ruler and compass? The Greeks could do 3, 4, 5, 6, 8, 10, 12, 15 and so on (doubling the sides is easy: just bisect). For two thousand years nobody found another. Then in 1796, an 18-year-old Carl Friedrich Gauss proved that the regular 17-gon is constructible. The story goes that he was so proud he asked for a 17-gon on his headstone. It never happened: the stonemason said nobody would be able to tell it from a circle, and the monument in Gauss's home town of Brunswick carries a 17-pointed star instead. (It is a much-repeated anecdote rather than a documented request, so treat it as a story.)

The full answer, completed by Pierre Wantzel in 1837, is: a regular n-gon is constructible exactly when n is a power of 2 times distinct Fermat primes. The known Fermat primes are 3, 5, 17, 257 and 65,537. So from 3 to 20 the constructible ones are 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, and the impossible ones are 7, 9, 11, 13, 14, 18, 19. The heptagon (7) and the nonagon (9 = 3 × 3, a repeated Fermat prime) cannot be drawn exactly.

Bisecting any angle is easy. So the ancient Greeks naturally asked: can you trisect any angle, cutting it into three equal parts, with only straightedge and compass?

For some angles, yes. A 90° angle trisects into 30° pieces, and we can construct 30°. A 180° angle trisects into 60° pieces. But the Greeks could never find a method that works for every angle, and in particular nobody could trisect 60° into three 20° pieces.

For over 2,000 years mathematicians kept trying. In 1837, the French mathematician Pierre Wantzel proved that it is impossible. Not just hard: impossible. No sequence of straightedge lines and compass circles, however long, can produce an exact 20° angle from nothing.

Here is a beautiful, complete answer. Among whole numbers of degrees, you can construct exactly the multiples of 3°, and nothing else.

Why every multiple of 3° is possible. You can construct 60° (equilateral triangle) and 72° (regular pentagon). Subtract: 72° − 60° = 12°. Bisect: 6°. Bisect again: 3°. Once you have 3°, you can copy it side by side as many times as you like: 6°, 9°, 12°, … every multiple of 3°.

Why nothing else is possible. If you could construct any whole-degree angle that is not a multiple of 3°, you could combine it with 3° (adding and subtracting copies) to get 1°. Then twenty copies of 1° would give 20°. But Wantzel proved 20° impossible. So 1°, 2°, 4°, 5°, 10°, 20°, 40°, 50°, 70°, 100°… are all out of reach.

These problems mix angle facts, constructions and careful reasoning. Try each before reading the solution. Olympiad problems reward a clear diagram and patient step-by-step thinking more than clever tricks.

A striking fact: modern CAD (computer-aided design) programs used by engineers are, underneath, digital versions of straightedge and compass. You pick points, draw lines through them and circles around them, and the program finds where they cross, exactly as Euclid did. A total station, the yellow instrument on a tripod you may see beside a new road, is a super-accurate protractor combined with a laser rangefinder. It measures angles to within a few seconds of arc (a second is 1/3,600 of a degree).