Given an equilateral triangle of side 185 mm, how do I calculate the diameter or radius of the circle that intersects with the three corners of the triangle?
My Zeus tables explain how to set up a drilling rig for a given diameter, but not the reverse.
The formula would be helpful.
A link to a web site of such useful information would be wonderful!
TIA
Richard
More thought and less haste is an obvious answer (with apologies).
In message , Richard writes
You mean like typing
"diameter of a circle on an equilateral triangle"
into google ?
which yields various site such as this ...
circumscribed radius of an equilateral triangle of side L = L * 0.5574
For 185mm side, this is 107mm
In message , R D S writes
plonker ...
I make it L/SQRT(3) = L/1.73205 (or L * 0.5774 not 0.5574) = 106.8 for L =
185 [I think that's probably what you meant, 'cos 185 * 0.5574 = 103 not 107]
If the equilateral triangle is converted into three equal sized triangles within the main one, using the centre of the circle as the common point where all three converge, you get three triangles with two 30deg angles and one of 120deg, and the lines from the side of the circle to the centre is the radius of the circle.
Half one of these triangle, you get a right angled triangle with a
30deg and 60 deg angles. Using trig, the 185mm triangle edge now = 2 x the (A)djacent side of the triangle in relation to the 30deg angle.Formula is therefore (185/2) / cos(30) = 106.80. As a generic formula, r = L/2 / cos(30).
Matt
You beat me to it :-)
I think it can be done (proofed) with no cos or sin involved IMHO.
Adam
Yes it can. I did it using Pythagoras and interesting chords. It comes out as r=L/sqrt(3) - which is the same thing anyway because cos(30)=sqrt(3)/2
My 1988 GCSE Maths didn't stretch to the relationships of Pythagoras and trig. Why does sqrt(3)/2 =3D cos(30deg)? I can see that it does, I just can't see the Pythagoras on the way to getting there......
Is it because of the defined relationship in a triangle with 30 / 60 /
90 degree angles (making the sides =3D 106.8, 53.4, 92.5 in this case?)I should have done A level maths (perhaps I will one day) - its fascintating stuff!
Matt
First off, a correction to my previous post. I've no idea what an "interesting" chord is - I of course meant "intersecting chords"!
To answer your question. . .
Imagine an equilateral triangle with side length 2 units. Draw a perpendicular from apex to base. You now have two 30/60/90 triangles with the 30 degree angle at the top. The hypotenuse is 2 units long and the side opposite the 30 degrees is 1 unit. Apply Pythagoras to that and (adjacent side)^2 = 2^2 - 1^2 = 4 - 1 = 3 The length of the adjacent side is thus sqrt(3) units. The cosine of 30 degrees - adjacent side/hypotenuse is therefore sqrt(3)/2
QED!
You say QED - but is it? Maybe being pedantic here but is this a proof as such? I mean what you have written is correct - but it's "just" a particular case that happens to give the result expected. As I say, not being pedantic but wonder if this really is a "proof"? I'm no mathematician but the definition of a cosine is, I believe, a series.
I'd say that if anything is missing it's that Roger didn't /prove/ that the perpendicular from one corner of the equilateral triangle to the opposite side actually bisects the angle and the opposite side. It's obvious from the symmetry that it does, but can you prove it starting only from Euclid's axioms?
Hmm... As usually taught you start with the simple right-angled triangle definitions - cos = adjacent/hypotenuse, etc. - and then extend that definition to allow angles outside the range 0 to 90 deg. Then the power series expansions can be derived and the trig functions generalised to allow complex number arguments. Hence if you can prove the bisection of the angle and side above and accept Pythagoras (which can be proved in several ways) you know that the 1 : root 3 : 2 triangle has angles of 30, 60 & 90 deg. and that cos 30 deg. = sqrt(3)/2, and so on.
This was second form stuff when I was at school. It's a bit worrying to read things like
(Discuss.)
Perhaps only worrying in that its now 20 years ago and (despite being an accountant) I left maths far behind me so it may simply be my forgetfulness.
Though I'm fairly sure that Pythagoras and trig were taught as sequential steps - first learn Pythagoras such that the dimensions of any right angled triangle can be worked out. Then learn trig in order that angles or lengths can be derived based on given information.
So whilst we all knew SOHCAHTOA etc, and we all knew Asqr =3D Bsqr + Csqr, the two formula were not merged into the (entirely logical as I see it now) position above.
That's not to say, though, that with a bit of thought and a simple explanation above I can't see how that works.
Perhaps what I should have written was "my 1998 Maths GCSE didn't stretch to being explicit about formulae which merged Pyth and trig, and instead kept them separate (but highly related of course) to make a two step process rather than a single step"
Matt
I think that is one of the few mathematical proofs I ever mastered! ;-)
I did GCE O level maths in '83 or '84, and don't recall trig getting much beyond basic right angle triangle stuff, with a possible requirement to be able to use the sine or cosine rules. I was not sufficiently a fan of maths (at the time) to take it at A level.
I do remember having quite a nasty shock when doing maths at university to find there was a whole world about trig identities and equivalents that I knew absolutely nothing about! (Much of the difficulty stemming from the fact that there were several maths groups one could be in, and I happened to get the one where 95% of the students who had done A level maths, and the very obtuse Welshman teaching, thought he could race through it all as a "quick bit of revision"). After a few weeks of that I decided to switch groups and found that the alternative was a running at a much more sensible pace (the first group had almost finished course in the first four weeks!)
It surprising really how even after having spent almost all of my working life in quite technical roles, just how little maths one is called upon to use.
Hmm - yes remember that one ;-)
(I think I preferred one of my maths teachers mnemonics of "Two Old Aunts Sat On High Chairs and Howled)
That (I discovered later in life) was one of the problems I often had with maths at that level - it was never really built from first principles and hence never really satisfied my desire to know *why* something worked.
A good example would be something like matrix operations. At O level there was never any worthwhile application given for why you might want to carry out these manipulations - which made them seem all rather pointless. Its only later when you realise you can convolve data sets, solve simultaneous equations, and do all sorts of fancy graphics with them (to name but a few applications) that you realise they do actually have a purpose.
John Rumm coughed up some electrons that declared:
The AO-level maths did cover a bit more of this sort of thing, also in 83 for me. A level maths covered a lot more and definately was the level physics at uni assumed.
Wave mechanics and associated maths blew my mind - never did get the hand of that stuff.
Never used any of it since, being a sysadmin and programmer.
I fused a braincell trying to do 3d-trig on my roof last month. Fortuanately the basics were still there but I had to work lots of things out the hard way, starting from the beginning.
Cheers
Tim
Silly Old Horse, Curley And Heavy, Trod On Albert
and " X equals minus B, plus or minus the square root of B squared minus
4 A C all over 2A" - Not used that in a very long time.There's nothing wrong with learning by rote. I can still multipy by reciting the tables, where necessary :-)
I was lucky in that I enjoyed doing my GCE O level and did go on to do an A level. I bumped into my A level teacher 20 years after doing my A level and took the time to tell her she was the best teacher I ever had (she was no longer teaching and was interviewing my wife for a job when I bumped into her). I have just dragged some of my old schoolbooks out of the loft to help a friends lad out who is starting A level next term.
She used to do maths things that were not on the course, just for fun. Learning that any recurring decimal can be written as a fraction is my favourite
eg .123123123 recurring or any other numbers (I have used 3 digits here but you can use more eg .1234512345) will always be a fraction if
(time to use letters for numbers and remember that these are recurring decimals)
.abcabcabc will be the fraction abc/999 .abcdabcdabcd will will be the fraction abcd/9999.
Adam
In message , ARWadworth writes
by far my best teacher at "O" level was "Nut" Hayes, who was B.Sc. Physics (Failed), stirred his tea with his comb etc ... but was prolly the best teacher I came across
Respect ... as one would say now
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