My Math Forum Constructing a 20 Degree Angle using Compass and Ruler Only

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 View Poll Results: Is this a useful construction? Yes 1 9.09% No 0 0% There are other ways 3 27.27% This method is wrong 7 63.64% Voters: 11. You may not vote on this poll

 January 14th, 2011, 10:57 PM #1 Newbie   Joined: Jan 2011 Posts: 2 Thanks: 0 Constructing a 20 Degree Angle using Compass and Ruler Only Q - Is it Possible to Construct a 20 Degree Angle using Compass and Straightedge only? Well, I have been through the web in this regard, and have not found any reliable method for construction. While in 6th Grade, I stumbled upon a highly accurate construction. I have posted my method of construction on my blogpage http://atharvjoshi.blogspot.com/2010...gle-using.html. I request you to please view it, discuss and if possible, are there are any other ways of this construction. Angle Trisection was thought to be a solution, but Angle Trisection is not possible. This elementary solution of mine may be the only method. Ahem! I will be posting the steps below: CONSTRUCTING A 20 DEGREE ANGLE USING COMPASS AND STRAIGHTEDGE (RULER) ONLY 1. Step 1: Draw a line segment PQ. With centre O, Draw an arc of any length using your compass and label it as AB. 2. Step 2: Keeping your compass wide open with the same length, draw an arc S on AB. This will be your normal 60 Degree angle arc. (If you join S to O, you will get an angle measuring 60 degrees i.e. Angle SOB). 3. Step 3: Bisect the 60 degree angle and name the point of intersection as J. J will be your normal 30 degree arc. (After bisecting the 60 degree angle, if you join J to O you will get an angle measuring 30 degrees i.e. Angle JOB). Till Step 3, it was a normal construction. Step 4 is where the 'Twist' occurs!!! 4. Step 4: Join the arc J to A, and there you have it! Angle JAB is your 20 degree angle! Now you can easily construct angles measuring 10,40,50,70,80,100,110,130,140,160,170 degrees! Thanks, Atharv Joshi
 January 15th, 2011, 03:37 AM #2 Global Moderator   Joined: Dec 2006 Posts: 18,166 Thanks: 1424 Angle JAB is less than 30°, but its value depends on where J is on the bisector of angle SOB, which your third step doesn't specify. It's impossible to construct an angle of exactly 20° using only classical ruler and compass construction methods. Last edited by skipjack; August 21st, 2017 at 07:10 AM.
 March 4th, 2011, 11:27 PM #3 Newbie   Joined: Jan 2011 Posts: 2 Thanks: 0 Re: Constructing a 20 Degree Angle using Compass and Ruler O @ Skipjack.... angle JAB will be 30 degrees, because you are bisecting the 60 degree angle SOB... I think the position of J will be the same no matter which compass length you take, provided that the compass length does not change right from step one. On your last sentence, Did you try the method out practically? It comes to 20 degrees..... coz I have tried it dozens of times.... otherwise I wouldn't post it on the internet.
March 6th, 2011, 10:02 PM   #4
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Quote:
 Originally Posted by atharvjoshi 3. Step 3: Bisect the 60 degree angle and name the point of intersection as J.
The point of intersection of what two lines?

Also, please explain why you posted both "angle JAB will be 30 degrees" and "Angle JAB is your 20 degree angle!"

If triangles OSB and JSB are distinct equilateral triangles, angle JAB = arctan $\displaystyle \frac{\sqrt{\small3}}{\small5}$ = 19.1066° approximately, not 20°.

Last edited by skipjack; August 21st, 2017 at 07:11 AM.

 April 12th, 2012, 12:05 PM #5 Newbie   Joined: Apr 2012 Posts: 2 Thanks: 0 Re: Constructing a 20 Degree Angle using Compass and Ruler O Me and my friend have constructed a 20 degree angle exactly and even our school teachers have approved; it we even have a proper justification. We want to upload it on internet keeping it safe plzz suggest. Contact 9987677085 Last edited by skipjack; February 18th, 2017 at 05:04 AM.
April 12th, 2012, 12:20 PM   #6
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by atharvjoshi angle JAB will be 30 degrees, because you are bisecting the 60 degree angle SOB... I think the position of J will be the same no matter which compass length you take, provided that the compass length does not change right from step one. On your last sentence, Did you try the method out practically? It comes to 20 degrees..... coz I have tried it dozens of times.... otherwise I wouldn't post it on the internet.
Are J, A, and B all on the same circle? Your construction implies that this is the case. If true then angle JAB = 15 degrees by basic high school geometry. I know this is true otherwise I wouldn't post it on the internet.

April 12th, 2012, 12:25 PM   #7
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by Gaurangsk Me and my friend have constructed a 2o degree angle exactly and even our school teachers have approved it we even have a proper jistification. We wAnt to upload it on internet keeping it safe plzz suggest. Contact 9987677085
I'm pretty certain that I constructed one too. I used a protractor and it came out exact.

On the other hand, if you mean that you constructed an angle of exactly 20 degrees using the classic Greek tools of a collapsible compass and unmarked straightedge, then save yourself and your teacher the humiliation. It can't be done.

 April 12th, 2012, 01:01 PM #8 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 10,927 Thanks: 718 Re: Constructing a 20 Degree Angle using Compass and Ruler O Atharvjoshi Gaurangsk Are you two related?
April 12th, 2012, 01:05 PM   #9
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by mrtwhs On the other hand, if you mean that you constructed an angle of exactly 20 degrees using the classic Greek tools of a collapsible compass and unmarked straightedge, then save yourself and your teacher the humiliation. It can't be done.
Sure it can: construct a regular 18-gon; draw 2 straight lines from center to 2 consecutive vertices

April 12th, 2012, 01:33 PM   #10
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Re: Constructing a 20 Degree Angle using Compass and Ruler O

Quote:
 Originally Posted by Denis Atharvjoshi Gaurangsk Are you two related?
The IP addresses don't match, FWIW

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