ArcTan without calculator


I had a situation yesterday that I would have solved by whipping out the DM15, but I had forgotten it, so manual calculation was the only option. I must though humbly admit that I utterly failed.
I wonder how I should have done.

We had a BBQ at a friend, who was constructing a new shed. The roof was not yet finished and we starting talking about roofing materials, but then came into that depending on the roof angle, there is a need for an underlayer. And then we wanted to know the roof angle...

We got as far as seeing it as a right angled triangle were the base is 3 and the height is 1, the side thus sqrt(10). Thus we only needed to calculate arctan(1/3) to know the roof angle. (I later put this to my neighbour who was a carpenter, he knew by heart several angles that you have when triangle height is 1 and base is an integer).

No calculators at the house. No smartphones. Nobody wanted to go in to boot up a computer.

Everyone pondered over the evening about how to solve this (series expansions: but who remembered the formula, integrals: something with 1 over a square root of something).

In the end, several calculations leading to nowhere, we had to give up and ring one of the kids who opened a Wolfram Alpha tab (what a computation overkill).

I don't know the arc trig or the regular trig expansions by heart, was there a way this could be solved logically with pre-calculus math?

Edited: 14 Aug 2012, 5:44 a.m.



1/ For 'small' angle in radian, ATAN(a)~=a

Pi ~= 22/7

so here :

  a ~ 180/3*7/22
~ 60*7/22
~ 420/22
~ 210/11
a ~ 19 °

If you take PI=3 (!) , just 180/9 -> ~ 20° not too bad

2/ With pen, paper and ruler ;)

Edited: 14 Aug 2012, 6:39 a.m.


1) It's so simple... Why couldn't we think of that !
2) A ruler that can measure degrees you mean (protractor)?

Edited: 14 Aug 2012, 6:41 a.m.


Even simpler :

ArcTan(1/3) ~> 1/3 ~> 0.3333 ~> PI/10 ~> 180°/10 ~> 18°

For point 2, i mean a 'rapporteur' but i don't know the english word

Edited: 14 Aug 2012, 12:16 p.m.


For point 2, i mean a 'rapporteur' but i don't know the english word

That's a "protractor" (as PatrickR mentioned)


To improve the accuracy you could use the following formula:

I assume you know how to calculate sqrt(10):

10 3
100 61
3900 626
14400 6322
175600 63242

Thus we get:

We end up with:

Compare that to the exact value: 18.4349

Obviously we don't really need four places.

Kind regards



Sure. x<<1| sin(x)~x, cos(X)~1, tan(x)~x => atan(x)~x
for atan(1/3) the error is less than 3.6%.


My first guess would be about twenty degrees. 1:sqrt(3):2 is a Pythagorean triad for a ninety/sixty/thirty degree triangle and this is a bit less.

- Pauli


A simple approximate solution is that from ratio 1:3 down, the angle is found by 57 (roughly 1 radian) divided by the dominator. E.g. 57/3=19, 57/4=14.25, 57/5=11.4, etc. The smaller the fraction, the more exact result.

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