/sci/ - Science & Math » Thread #13887968

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Anonymous Thu 25 Nov 01:38:25 2021 No.13887968 View Reply Original Report

Quoted By: >>13888011 >>13888498 >>13888574

How would you measure the narrowness of a triangle?

Clearly an isosceles is narrower than an equilateral with the same base; or is it? What if the bases are different?

Anonymous

Anonymous Thu 25 Nov 2021 01:56:50 No.13888006 Report

Quoted By:

the smallest angle, pretty straightforward lad

Anonymous

Anonymous Thu 25 Nov 2021 01:59:49 No.13888011 Report

Quoted By:

>>13887968
smallest a + b - c where a,b,c are side lengths

Anonymous

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Anonymous Thu 25 Nov 2021 06:10:13 No.13888498 Report

Quoted By: >>13888506 >>13888590

>>13887968
I was playing with the idea of a "Euclidean angle"
as a way to conveniently judge the narrowness
of angles and other neat properties.

The angle is basically the length of BC divided
by the length of AD. This gives the angle of A.
If BC is long or AD is short, the angle is broad
and vice versa. By this definition, the product
of two angles always equals 4 when they
are supplementary, and the sum of two angles
always equals 2 when complementary.

What do you think, OP?

Anonymous

Anonymous Thu 25 Nov 2021 06:12:16 No.13888506 Report

Quoted By: >>13888590

>>13888498
that's tan(A/2)

Anonymous

Anonymous Thu 25 Nov 2021 06:42:35 No.13888574 Report

Quoted By:

>>13887968
the ratio of ab to cb + ca

Anonymous

Anonymous Thu 25 Nov 2021 06:54:24 No.13888590 Report

Quoted By:

>>13888498
>>13888506
Yes, that's true. I just remapped that tangent and
made the "angle" run from 0 to infinity. In the
absence of a protractor, one can use the "angle"
to describe how wide it is horizontally and how
long it is vertically to reliably and practically
make it.

Anonymous

Anonymous Thu 25 Nov 2021 08:19:12 No.13888697 Report

Quoted By:

Well its kind of a weird question. In one sense you could be asking "how thin a channel can this triangle fit through?" That would mean assuming the base is the longest side and finding the height from there. This does nothing to capture the difference between two triangles who can fit through the same channel, but which may have arbitrarily different base lengths. Measuring the smallest angle doesn't really work since one can keep a base length and height the same while shifting the top of an isosceles triangle on a line which maintains its height but changes the angles - you'll get different measurements for 'narrowness' using smallest angle even though the channel it can fit through and its longest side need not change.
To get around those things, I suggest you take the ratio of the longest side to the height. This has a partly ugly property that an equilateral triangle will give you a narrowness measure of root{3}/2 rather than 1, but it doesn't get confused for triangles of different sizes since it is but a ratio.

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