# Thread: Finding maximum area of triangle

1. ## Finding maximum area of triangle

Hi, I have the following problem.

"An isosceles triangle ABC, where AB = AC and BāC = 2(alpha) is inscribed in a circle of radius r. Show that the area of the triangle ABC is a maximum if alpha = pi/6."

To be honest I have no idea where to start. Also, I'm not sure if I'm in the right subforum, so move me if need be.

2. ## Re: Finding maximum area of triangle

Well, I know at least that I'm looking for dA/d(alpha), which is equal to dA/dr * dr/d(alpha)
dA/dr would be the derivative of the area respect to r, but I'm not sure what the formula for the area of this triangle is.. heheh..
dr/d(alpha) I -think- I can get with the cosine rule, from where I get the relationship arccos((a^2 -2r^2)/(-2r^2)) = alpha.
Derivating that I get dr/da = $-2*a/(r*sqrt(2*r-a)*sqrt(2*r+a))$

So yup, any help would be appreciated.

3. ## Re: Finding maximum area of triangle

Wikipedia has the following formula for area: $S=2r^2\sin(2\alpha)\sin^2\beta$ where $\beta$ is the measure of the other two angles.

4. ## Re: Finding maximum area of triangle

The following link might be useful
Isosceles Triangle Equations Formulas Calculator - Area Geometry

Has many useful formulas for various triangles. One of which is the radius of a circumscribed circle
$r=\frac{a}{2 \sin(A)}$

Switching it around you get
$\sin(A)=\frac{a}{2 r}$

You can plug those into the area formula and go from there.

5. ## Re: Finding maximum area of triangle

Recall that in any triangle, $\sin{A}=\frac{2\Delta}{bc}$. " $\Delta$" is the area of the trianlgle.
If k is one of the equal sides, $\sin{2\alpha}=\frac{2\Delta}{k^2}$.........[1]

Also, $\cos{(\pi-2\alpha)}=\frac{2r^2-k^2}{2r^2}$.........[Cosine Formula in $\Delta AOC$]

$-\cos{(2\alpha)}=\frac{2r^2-k^2}{2r^2}$

$k^2=2r^2[1+\cos{2\alpha}]$.........[2]

Substitute [2] into [1]:
$\sin{2\alpha}=\frac{2\Delta}{2r^2[1+\cos{2\alpha}]}$

This is the relation between $\alpha$ and area of the triangle.
$\Delta = r^2[\sin{2\alpha}+ \sin{2\alpha}\cos{2\alpha}]$

$\frac{d\Delta}{d\alpha} = r^2[4\cos^2{\alpha}(2\cos{2\alpha}-1)]$

I hope you can easily proceed from here...