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I wanted to know if it is possible to construct a $72^{\circ}$ angle via a ruler and compass? I know some numbers cannot be constructed using a ruler and compass, due to Galois Theory, but I don't exactly know how to check if a number falls in that category or not. Any resources would be appreciated as well!

Martin Sleziak
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Aditya
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    Since you can construct a regular pentagon (and its exterior angle is what you need), why not? https://en.wikipedia.org/wiki/Pentagon#Construction_of_a_regular_pentagon – Deepak Aug 22 '21 at 06:00
  • To answer your broader question, think about what regular polygons are constructible, and what angles are derivable from them. Constructible polygons are intimately related to Fermat primes, – Deepak Aug 22 '21 at 06:03
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    $72^o=360^o/5$, See the construction of the pentagon. – Vasile Aug 22 '21 at 07:31
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    Does this answer your question? Construction of a regular pentagon – Nij Aug 22 '21 at 07:56
  • You can calculate $\cos72^{\circ}$, and then you can use Galois Theory to show that number is constructible, then you can use that construction to get a $72^{\circ}$ angle. – Gerry Myerson Aug 22 '21 at 08:56

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This doesn't need Galois theory. We have $\cos(72^\circ)=\frac{\sqrt5-1}4$, so we can construct $72^\circ$ using the following steps:

  1. Draw a line of length $\sqrt5$. (It is well-known that you can construct $\sqrt a$ from $a$)
  2. Subtract $1$.
  3. Divide by $4$. (Repeatedly bisecting; this creates $\cos(72^\circ)$)
  4. Draw a right triangle with hypotenuse $1$ and adjacent $\cos(72^\circ)$. (In detail: Draw a perpendicular line from the line segment of length $\cos(72^\circ)$ drawn above, and intersect with a circle of length $1$.)
  5. The angle the two lines above create is $72^\circ$.
Kenta S
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