Finding Pi

[PhilOfPerth]

Or, if I read the problem wrong (as it seems Pete and I have two different interpertations of the problem), you might want:

Formula for the Base of an Isosceles Triangle
If you know the side length and height of an isosceles triangle, you can find the base of the triangle using this formula:

b = SQR(a ^ 2 - h ^ 2), where a is the length of a side and h is the height.

how to find the third side of a triangle: formula for the base of an isosceles triangle

where a is the length of one of the two known, equivalent sides of the isosceles.

But we don't know the height - it depends on two things, the side length and (half of) the base length (or the top angle), and the base length is what we want to find.

[PhilOfPerth]

@PhilOfPerth are you attempting this:
https://www.google.com/search?client=ope...Pl-SbaA_70

Yes! Looks like I'm a cupla hundred years late!
The Archimedes method is exactly what I'm trying to do, but I've never heard of it before!
I had drawn 3,6,8 and 12-sided shapes inside a circle but could only get the lengths of the sides using trig, which I wanted to avoid.
Thanks both for your help!

[bplus]

Well sir, if you are thinking like Archimedes it's a very good sign! That guy anticipated calculus with his Method of Exhaustion.

[jcm]

C=2*A*SIN(T/2)

[PhilOfPerth]

Well sir, if you are thinking like Archimedes it's a very good sign! That guy anticipated calculus with his Method of Exhaustion.

The difference is, he did it - and without this Forum for help - and I still can't do it!
I'm good at times-tables, but. 

[Pete]

Oh he came to the forum often, only back then it was called the QBasic Forum.

Pete

[SMcNeill]

I'm good at times-tables, but. 

Times-tables butt?   I'm good at that too!  That's when you have multiple strippers come over to a table where you and multiple buddies of yours are sitting, and you have to sort out who is shoving their dollar bills where so that you keep the dancers' interest, without overlapping payment and using up your beer money too quickly.

I'm an expert at times-table butt!!

[bplus]

One for the Math (or ”outside-the-box-thinking”) gurus:  
Given an isoscles triangle ABC, with sides b and c (the sides opposite B and C) both 5 units in length, and with angle A=45degrees, is there a (simple?) way to find the length of side a, without resorting to pre-determined trig tables like sin, cos and tan, or pi? 
The reason I don’t want to use these is I’m trying to demonstrate how pi relates to the circumference of a circle, so I don’t want to involve anything that relies on pi – that would be “bootstrapping”,  sort of like lifting oneself up by the bootlaces.

Trig tables do not rely on Pi. 

Sin, Cos, Tan... are Constant Ratios of various sides of a Right Triangle at various angles.
And Pi itself is a Constant Ratio of Radius length to 1/2 circumference, no matter the length of radius.

For example Sin of an angle = the length of the side opposite that angle divided by the length of hypotenuse of that right triangle

[bplus]

C=2*A*SIN(T/2)

Welcome @jcm to forum! Care to elaborate on A, C and T? A and C might be sides but using Phil's example being capital they would be Angles as sides get lower case letters by convention. Is that formula from a Trig Identity?