# Thread: What is the largest area of an isoceles triangle with

1. ## What is the largest area of an isoceles triangle with

What is the largest area of an isosceles triangle with the perimeter 1 cm?

2. Hello,

I find $\displaystyle \color{red}\frac{\sqrt{3}}{36}$ cm.

I give my solution after diner.

3. Originally Posted by gtgamer140@yahoo.com
What is the largest area of an isosceles triangle with the perimeter 1 cm?
Hi

Let a be the equal length of 2 sides and b the length of the 3rd side.

The perimeter is p = 1 = 2a+b

The are is given by the formula $\displaystyle S = \sqrt{\frac{p}{2}\left(\frac{p}{2}-a\right)\left(\frac{p}{2}-a\right)\left(\frac{p}{2}-b\right)}$

$\displaystyle S = \sqrt{\frac{1}{2}\left(\frac{1}{2}-\frac{1-b}{2}\right)\left(\frac{1}{2}-\frac{1-b}{2}\right)\left(\frac{1}{2}-b\right)}$

$\displaystyle S = \sqrt{\frac{1}{2}\:\frac{b}{2}\:\frac{b}{2}\:\frac {1-2b}{2}}$

$\displaystyle S = \frac{b}{4}\:\sqrt{1-2b}$

Set the derivative of S to 0 to find b_max then S_max

4. Exactly,

But it's not necessary to know this formula.

You have $\displaystyle h^2=a^2-\left(\frac{b}{2}\right)^2$ with $\displaystyle b=1-2a$ and $\displaystyle S=\frac{bh}{2}$.