Finding a solution for non-linear simultaneous equations

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• March 5th 2012, 06:21 AM
malby
Finding a solution for non-linear simultaneous equations
Hi

I am not sure if the title accurately reflects my issue, apologies if this is not the case.

I am an academic engineer, I have encountered a problem where i would like to equate a circle to a triangle to simplfy a problem..... My criteria is to transform a circle to a 120, 30, 30 triangle with equivalent area and perimeter, when i quickly derived the simultaeous equations i get the following two expressions

y1 = b+2*h
y2=(b*h)/2

where:
y1=the circles perimeter
y2=the circles area
b= base of triangle
h= height of triangle

I thought i would ask as i do not want to waste time if it is not possible to solve and satisfy the 120, 30, 30 condition......... Are there any methods that can be used to tackle this issue?
• March 5th 2012, 11:15 AM
Deveno
Re: Finding a solution for non-linear simultaneous equations
if b,h are given, $y_1$ and $y_2$ may simply be calculated. so i assume that $y_1, y_2$ are given, and you wish to find b and h.

write: $b = y_1 - 2h = \frac{2y_2}{h}$

from $y_1 - 2h = \frac{2y_2}{h}$ we get:

$2h^2 - y_1h - 2y_2 = 0$, which is a quadratic in h.

the discriminant is $y_1^2 - 8y_2$

however, $y_1,y_2$ aren't "independent" they both depend on the radius of the circle, r:

$y_1 = 2\pi r$
$y_2 = \pi r^2$

so $y_1^2 - 8y_2 = (2\pi r)^2 - 8\pi r^2 = -4\pi r^2 < 0$, therefore: no real solutions for h.

however, in reviewing your original equations, it appears that your formula for either the area or the perimeter of the isoceles triangle is incorrect, if h is the height, then the perimeter length should be:

$b + 2(\sqrt{h^2 + \frac{b^2}{4}})$.

you might want to review that, and confirm (the calcuations get a lot messier if this is true).
• March 6th 2012, 04:10 AM
malby
Re: Finding a solution for non-linear simultaneous equations
Thanks, I did make a couple of mistakes actually, sorry about that, the equations should be

y1 = b+2*(sqrt(b^2+h^2))
y2=(b*h)/2

where:
y1=the circle z's perimeter (GIVEN)
y2=the circle z's area (GIVEN)
b= base of triangle
h= height of triangle

I want to calculate values for b, and h for a 120, 30, 30 triangle, my main question is there a solution? i.e. can you take a circle and construct a 120, 30, 30 triangle with an identicle Area and perimeter?