Finding Pi

[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