Split a straight angle into ten equal angles. What is the measure of each smaller angle?
Answers
Answer:
the smaller angle is called as acute angle
Well the answer is clearly pi/10 radians = 18 degrees. Doing the division using compasses and a straight edge is a different matter altogether.
It is straightforward to divide 180 degrees into two right angles using a straight edge and compasses so the problem reduces to dividing a right angle into five equal parts.
A method for doing this was given by Sivaram on Quora (24th Nov 2017). Unfortunately this method does not work. I have calculated the first angle using this method which turns out to be arctan(1/4) or 14.036 degrees
The problem of dividing pi/2 into 5 equal parts is equivalent to constructing a 20 sided regular polygon using straight edge and compasses because the “turning angle” at each vertex is pi/10 radians. A classic theorem on the constructability of regular polygons states that a reqular n sided polygon can be constructed if n is the product of a power of two and any combination of distinct Fermat primes. Since 20 = 2^2 x5 and 5 is a Fermat prime the construction is possible.
Iooking at You tube I have found videos showing how to construct regular pentagons and decagons inscribed in a circle. given the construction of a decagon, the construction of a regular 20 sided polygon can be achieved simply by bisecting each side.
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