## Saturday, May 21, 2011

### Constructing a 20 Degree Angle using Ruler and Compass

Ahoy there readers! This can be labelled as the first 'constructive' post of my blog. Yes Indeed, it is literally "CONSTRUCTIVE".

I will be showing you an extremely simple method to Draw a 20-Degree Angle using Ruler and Compass only in 4 steps. This method is my own, and I came across it sometime in 6th grade when I 'misconstructed' a 30-degree angle. Till now, after hours of searching, I have yet not found a proper method of constructing a 20-degree angle with ruler and compass on the internet. Before revealing the method, I will try to give an introduction to Compass and Straightedge construction. Before beginning, let me make it clear that I am no mathematician, and you are free to criticize any aspect of this post.

Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angle, and other geometric figures using only a ruler and compass. This was the classical Greek way of Geometry. But the Greeks did not find constructions for three problems, one of them being Angle Trisection. Angle Trisection is still considered impossible, and so is constructing a 20 degree angle with compass and straightedge.

Angles that can be constructed by Compass and Straightedge are 15,30,45,60,90,120,150,180 degrees and some other angles can be constructed by Bisection (for eg. 7.5 degrees, 22.5 degrees e.t.c). Constructing a 20 degree angle can help you in constructing 10,40,50,70,80,100,110,130,140,160,170 degree angles with only a Ruler and Compass (and your pencil)!

Given below is my method of carrying out the construction of a 20 degree angle using a straightedge and compass. It is very accurate and you get a perfect measure everytime! Readers, feel free to criticize, point out mistakes and also, don't forget to Praise! And hey, also make an effort to see if this construction had been carried out before (and comment below, please)!

The figures below are my own work, and are not perfect drawings, and the image is only a representation, not an actual geometric construction.

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 an
d 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!

PS. If you still didn't understand the construction, or have any queries, mail me at atharvjoshi@ymail.com.

Later!

1. good job :-)

2. Nice post dude!!!!!!!!!!!

3. Sorry Dear, But I Have to say That this is not correct 20 degree angle. you have made 19 degree angle by this method.
Regards ASHISH KHANNA

4. Yes, more exactly, its 19.11 degrees. Great job though, pretty close.

5. @ The latter 2 anonymous posters....

19.11???? How come??? can u plz explain.... coz I am just a 13 year old! and I can't see any difference between this 19.11 and 20 degrees on my normal geometry box calculator..... If i would have known it was 19.11 degrees, I wouldn;'t have posted it in the first place :(

6. Dude.. pls teach us how to draw a 70 degree angle ok.. eager to see ur talent on this..

7. Dear AAJ,

Start by assuming you have a radius of one (doesn't really matter what number you choose, but "1" makes the calculations a bit easier).

Drop a perpendicular from J to line AB. Call this point H. Then triangle AHJ is a right triangle. It shouldn't be too hard to figure out that BH is 1/2. And so AH has a length of 5/2. Using the Pythagorean theorem, JH has a length of sqrt(3)/2.

Using trigonometry, the tangent of angle JAH is the length of JH divided by AH. That's sqrt(3)/5 which is approximately .3464. If you have access to a table of trig values, look this up under tangent and you get 19 degrees.

If you have a calculator with an atan function on it, enter atan(sqrt(3)/5) to get a better approximation (though you'll need to make sure your calculator is set for degrees instead of radians).

I hope this helps. I check this blog in a few days to see if you have any questions.

Professor E.

8. Hey.. its a gr8 way.. when i was in 7th i founded how to construct 20 degree but it was too complicated. I think this is the simplified version for the same. Thank you, for such a method

9. wow!don't worry for this time @@J, i'm sure we'll have an invention/discovery or 2 from you in the coming years:)
But to say it was a great method that you stumbled upon.
Can have a high middle school teacher starstruck ;)(except if she knows log tables)

10. THIS IS GOOD BUT DRAW A 79 DEGREE OR OR OR LOL

11. Hmm.... then the plus point is that we now can make angles which are multiples of 19.11 degrees! Cool!

12. i will surely try...

13. i can construct angle of any degree by using only compass. if anybody wanted to know contact me 7404140309

14. all those who think in is coming 19 degree or something are idiots i tried it 26 times but still it is coming 20 degrees.Try it again friends he is correct.

15. hey man its the wrong method.....idiot

16. purnima bansal, you are retarded. whats the right method then? f tard

17. HA! HA! HA! SOLVED MY PROBLEM BUT IT IS KINDLY REQUESTED TO PLEASE PROVE THIS THROUGH GEOMETRIC METHOD.

18. sorry but u have constructed 1.5/4π degree.

19. 19.11 degree angle it is actually impossible to draw a 20 degree angle sing just a ruler and compass does anyone kow why?

20. Hi. Bansal ji ediot kisi ko mat kahiye. It may be true method because instrumemt may be wrong.

21. 40 degrees??

22. well it is good, but gives 19.11 degree....lost 1/2 mark for my exams >.<
well, i still appreciate it for trying, you rock bro!!

23. A very simple and more accurate method for a 20 degree-angle construction ( 20,003.. degree) using ruler and compas only is given on page 3 in the paper :
"tracé d'un angle quelconque à la règle et au compas"
http://www.scribd.com/JJacquelin/documents

24. i too have made a 20 degree angle its accurate and has proper justification also

25. As you may have found out already, the construction 0.89 degrees wrong.
I would advise you to stop searching for a method since it has been shown impossible using galios theory

27. I have also constructed a fully 100% accurate 20 degree angle by mistake!!! I don't now remember the measurements but only the steps as it can only be made in sone range of measurements (not outside it but not only on one measurement)

28. sorry it was not only on one measurement, there is a range of them!!!

29. You constructed 19,11 degree angle. Look at the image for proof.
Igor, math teacher

30. how to construct a special angle 45 degree using compass and ruler

31. You cannot construct 20 degree angles with a compass and ruler. The only constructable angles by compass and are ruler and angles (integer valued) that are divisible by 3.

32. This is a consequence of the theorem that states if a is a constructable number then the rational field extended by a over the rational field is a power of 2. There is a proof that shows that 20 degrees is the solution to a minimal polynomial of power 3 which means the rational field extended by 20 degrees is 3 which cannot be a power of 2. So 20 degrees is non constructable.

33. What is the theorem apply to construct the angle 20
degree