How do you make 40 degree angle using compass and ruler what is the maths behind it?

Answer 1

Using a protractor rather than a compass is easier as follows:-

Draw a straight line of about 8 cm then place a protractor onto the line with zero degrees at the beginning of the line then mark off 40 degrees.

Remove the protractor and join the mark to the beginning of the straight line and 'hey presto!' you have constructed a 40 degree acute angle.

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It isn't possible to construct a 40 degree angle with only a straightedge and compass. There are well-known constructions for making a 60 degree angle and a 90 degree angle. Building on this, you can make an angle of any multiple of 15 degrees by adding and/or subtracting these angles. If you successively bisect these angles, you get angles of 30, 15, 7.5, 3.75, ... and 45, 22.5, 11.25, ... etc. degrees. Unfortunately there is no way to combine these angles to get exactly 40 degrees.

If there were a way to trisect an angle, this would open up many more possibilities. For instance if you trisect a 120 degree angle, you get 40 degrees.

In traditional geometrical constructions, you are only allowed to use a ruler and straightedge - a ruler with no marks on it. The Greeks made this rule several centuries BC. For at least 2500 years people have tried to find a way to trisect an angle, but no one has succeeded.

If you change the word "straightedge" to "ruler", (all you really need to do is make a pencil mark on the straightedge), then there is a method, but it isn't considered a construction, and it's not really accurate because you have to juggle the position of the ruler until it fits in a certain position.